লগ ইননিবন্ধন
বিষয়সমূহ

PrepBank · বিষয়ভিত্তিক প্রশ্ন

Number System, Problems on Number

মোট প্রশ্ন১,৭৩৬এই পাতা১০০প্রতি পাতা১০০
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

Number System, Problems on Number

PrepBank · পাতা ১৩ / ১৮ · ১,২০১–১,৩০০ / ১,৭৩৬

১,২০১.
The number of two digit prime numbers which remain prime even inverting the position of its digits is:
  1. ক
    4
  2. খ
    5
  3. গ
    9
  4. ঘ
    10
ব্যাখ্যা
Question: The number of two digit prime numbers which remain prime even inverting the position of its digits is:

Solution:
These numbers are 11, 13, 31, 17, 71, 37, 73, 79, 97.

∴ There are 9 such number.
১,২০২.
What is the least perfect square that is a multiple of 7, 11 and 12?
  1. ক
    ক) 421231
  2. খ
    খ) 242131
  3. গ
    গ) 223121
  4. ঘ
    ঘ) 213444
ব্যাখ্যা
Let us assume the least perfect square be X
⇒ 7 = 7 × 1
⇒ 11 = 11 × 1
⇒ 12 = 22 × 3 ⇒

The LCM of (7, 11, 12) = 22 × 3 × 11 × 7
⇒ The least perfect square = 22 × 32 × 112 × 72 = 213444
∴ The required result will be 213444.
১,২০৩.
If A381 is divisible by 11, find the value of the smallest natural number A?
  1. ক
    5
  2. খ
    6
  3. গ
    7
  4. ঘ
    9
ব্যাখ্যা
Question: If A381 is divisible by 11, find the value of the smallest natural number A?

Solution:
A number is divisible by 11 if the difference of the sum of the digits in the odd places and sum of the digits in even place is zero or divisible by 11.
Hence, (A + 8) - (3 + 1) = 0 or multiple of 11.
To get the difference 0 or multiple of 11, we need 7 at the place of A.
So, sum of odd place - sum of even place
= 15 - 4 = 11. And this is divisible by 11.
১,২০৪.
A worker was hired for 7 days. Each day, he was paid Tk. 10 more than what he was paid for the previous day of work. the total amount he was paid in the first 4 days of work equaled the total amount he was paid in the last 3 days. What was his staring pay?
  1. ক
    ক) Tk. 90
  2. খ
    খ) Tk. 138
  3. গ
    গ) Tk. 150
  4. ঘ
    ঘ) Tk. 160
ব্যাখ্যা
Question: A worker was hired for 7 days. Each day, he was paid Tk. 10 more than what he was paid for the previous day of work. the total amount he was paid in the first 4 days of work equaled the total amount he was paid in the last 3 days. What was his staring pay?

Solution: 
ধরি,
starting payment ছিলো x টাকা

২য় দিন = x + 10
৩য় দিন = ( x + 20)
৪র্থ দিন = ( x + 30)
৫ম দিন = ( x + 40)
৬ষ্ঠ দিন = ( x + 50) 
৭ম দিন = ( x + 60) টাকা

প্রশ্নমতে,
x + (x + 10) + (x + 20) + (x + 30) = (x + 40) + (x + 50) + (x + 60)
4x + 60 = 3x + 150
4x - 3x = 150 - 60
x = 90
১,২০৫.
What is the L.C.M of the numbers 36, 54 and 90? 
  1. ক
    ক) 18
  2. খ
    খ) 120
  3. গ
    গ) 356
  4. ঘ
    ঘ) 540
ব্যাখ্যা
Question: What is the L.C.M of the numbers 36, 54 and 90? 

Solution: 
36 = 2 × 3 × 2 × 3
54 = 3 × 3 × 3 × 2
90 = 2 × 3 × 3 × 5 

∴ the L.C.M of the numbers 36, 54 and 90 is = 2 × 2 × 3 × 3 × 5 × 3
= 540
১,২০৬.
How many '8' will you pass on the way when you count from 1 to 100?
  1. ক
    ক) 11
  2. খ
    খ) 20
  3. গ
    গ) 80
  4. ঘ
    ঘ) 70
ব্যাখ্যা
Question: How many '8' will you pass on the way when you count from 1 to 100?

Solution:
১ -১০ পর্যন্ত ৮ আছে = ১ টি
১১ - ২০ পর্যন্ত ৮ আছে = ১ টি
২১ - ৩০ পর্যন্ত ৮ আছে = ১ টি 
৩১ - ৪০ পর্যন্ত ৮ আছে = ১ টি
৪১ - ৫০ পর্যন্ত ৮ আছে = ১ টি
৫১ - ৬০ পর্যন্ত ৮ আছে = ১ টি
৬১ - ৭০ পর্যন্ত ৮ আছে = ১ টি
৭১ - ৮০ পর্যন্ত ৮ আছে = ২ টি
৮১ - ৯০ পর্যন্ত ৮ আছে = ১০ টি
৯১ - ১০০ পর্যন্ত ৮ আছে = ১ টি

∴ মোট ৮ রয়েছে = ২০ টি
১,২০৭.
Two numbers are in the ratio 3 : 5. If 9 is subtracted from each, the new numbers are in the ratio 12 : 23. The smaller number is
  1. ক
    ক) 27
  2. খ
    খ) 33
  3. গ
    গ) 49
  4. ঘ
    ঘ) 55
ব্যাখ্যা

ধরি,
ক্ষুদ্রতর সংখ্যাটি 3x এবং বৃহত্তর সংখ্যাটি 5x
প্রশ্নমতে, (3x - 9)/(5x - 9) = 12/23
⇒ 69x - 207 = 60x - 108
⇒ 69x - 60x = 207 - 108
⇒ 9x = 99
⇒ x = 11
অতএব, ক্ষুদ্রতর সংখ্যাটি = 3 × 11 = 33

১,২০৮.
(.1 × .01 × .002)/(.2 × .02 × .002) = ?
  1. ক
    0.0001
  2. খ
    0.25
  3. গ
    .50
  4. ঘ
    0.625
ব্যাখ্যা
Question (.1 × .01 × .002)/(.2 × .02 × .002) = ?

Solution: 
(.1 × .01 × .002)/(.2 × .02 × .002)
= 0.000002/0.000008
= 2/8
= 1/4
= 0.25
১,২০৯.
Let N be the smallest positive integer that is divisible by both 18 and 24. How many distinct prime factors does N have?
  1. ক
    2
  2. খ
    3
  3. গ
    5
  4. ঘ
    6
ব্যাখ্যা

Question: Let N be the smallest positive integer that is divisible by both 18 and 24. How many distinct prime factors does N have?

Solution:
এখানে, N হলো 18 এবং 24 দ্বারা বিভাজ্য ক্ষুদ্রতম সংখ্যা।
সুতরাং, N হবে 18 এবং 24 এর ল.সা.গু।

এখন, 18 = 2 × 3 × 3 = 21 × 32
এবং 24 = 2 × 2 × 2 × 3 = 23 × 31

LCM(18, 24) = 23 × 32 = 8 × 9 = 72
অতএব, N = 72

72 এর মৌলিক উৎপাদক = 23 × 32

স্বতন্ত্র মৌলিক উৎপাদকগুলি হলো 2 এবং 3।

∴ N এর স্বতন্ত্র মৌলিক উৎপাদকের সংখ্যা হলো 2টি।

১,২১০.
Which of the following is a rational number?
  1. ক
    ক) √3 × √9
  2. খ
    খ) √2 × √4
  3. গ
    গ) √2 × √8
  4. ঘ
    ঘ) √2 × √9
ব্যাখ্যা
Question: Which of the following is a rational number?

Solution:
We know,
Rational number × Irrational number = Irrational Number.
Here,
√3 × √9 = √3 × 3 = 3√3 is a irrational number.
√2 × √4 = 2√2 is a irrational number.
√2 × √8 = √16 = 4  is a rational number.
√2 × √9 = √2 × 3 = 3√2 is a irrational number.
১,২১১.
Three numbers are in the ratio 4 : 5 : 6 and their average is 25. The largest number is:
  1. ক
    30
  2. খ
    42
  3. গ
    32
  4. ঘ
    36
ব্যাখ্যা
Question; Three numbers are in the ratio 4 : 5 : 6 and their average is 25. The largest number is:

Solution: 
Let the three numbers be 4x, 5x and 6x

According to the question,
(4x + 5x + 6x) /3 = 25
⇒ 15x = 25 × 3
⇒ 15x = 75 
⇒ x = 75/15 =5 

largest number = 6x = 6 × 5 = 30
১,২১২.
If 'a' is an odd number and 'b' is an even number, which one of the following must be an even number?
  1. ক
    a2 + b2
  2. খ
    ab + 1
  3. গ
    a + b
  4. ঘ
    (a + b)2 + 1
ব্যাখ্যা
Question: If 'a' is an odd number and 'b' is an even number, which one of the following must be an even number?

Solution:
Suppose, a = 1 and b = 2

a + b = 1 + 2 = 3
ab + 1 = 1 × 2 + 1 = 3
a2 + b2 = 12 + 22 = 5
(a + b)2 + 1 = (1 + 2)2 + 1 = 10
১,২১৩.
The product of two numbers is 72 and the sum of their squares is 145. The sum of the numbers is:
  1. ক
    7
  2. খ
    8
  3. গ
    13
  4. ঘ
    17
ব্যাখ্যা
Question: The product of two numbers is 72 and the sum of their squares is 145. The sum of the numbers is:

Solution: 
let the two numbers be x, y 

xy = 72 
x2 + y2 = 145
⇒ (x + y)2 - 2 × 72 = 145
⇒ (x + y)2 = 289
∴ x + y = 17
১,২১৪.
equals how many eighteenths?
  1. ক
    12
  2. খ
    14
  3. গ
    18
  4. ঘ
    15
ব্যাখ্যা
Question:  equals how many eighteenths?

Solution: 
1/2 + 1/3
= 5/6

now,
( 5/6 ) ÷ ( 1/18 )
= 15
১,২১৫.
Two positive numbers are in the ratio 3 : 2. The product of their HCF and LCM is 3456. Find the sum of both the numbers. 
  1. ক
    186
  2. খ
    120
  3. গ
    144
  4. ঘ
    None of the above
ব্যাখ্যা

Question: Two positive numbers are in the ratio 3 : 2. The product of their HCF and LCM is 3456. Find the sum of both the numbers.

Solution:
Let two numbers are 3a and 2a.

HCF × LCM = 1st no. × 2nd no.
⇒ 3456 = 3a × 2a
⇒ 3456 = 6a2
⇒ a2 = 576
∴ a = 24 

∴ Sum of both the numbers = 3a + 2a = 5a = 5 × 24 = 120

১,২১৬.
Which of the following fractions is the largest?
  1. ক
    3/2
  2. খ
    7/4
  3. গ
    5/3
  4. ঘ
    6/5
ব্যাখ্যা
Question: Which of the following fractions is the largest? 

Solution: 
3/2 = 1.5
7/4 = 1.75
5/3 = 1.66
6/5 = 1.2 
Hence the largest fraction is 7/4 
১,২১৭.
A number x is 32% of a number y. If y is 20% of z, what is z in terms of x?
  1. ক
    0.064x
  2. খ
    0.64x
  3. গ
    6.4x
  4. ঘ
    15.625x
ব্যাখ্যা
Question: A number x is 32% of a number y. If y is 20% of z, what is z in terms of x?

Solution:
x = (32/100) × y 
⇒ x = (32y)/100
∴ 100x = 32y ..............(1)

y = (20/100) × z
⇒ y = (20z)/100 
∴ y = z/5

Put the value of y in equation (1),
100x = 32 × (z/5)
⇒ 100x = (32z)/5
⇒ 500x = 32z
⇒ z = 500x/32
⇒ z = 15.625x
১,২১৮.
The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?
  1. ক
    1
  2. খ
    10
  3. গ
    11
  4. ঘ
    19
  5. ঙ
    None
ব্যাখ্যা
Question: The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?

Solution:
Mean of numbers = 0
∴ sum of 20 numbers = 0 × 20 =0
It is possible that 19 of these numbers may be positive and their sum is a , the 20th number is (-a).

Means of 20 numbers would be = (n1 + n2 + ... + n20)/20  =0
Now we take at most case -
let us assume n1, n2, ...n19 are greater then 0
then
n20 = - (n1 + n2 +...+ n19)
Hence, in at most case there are 19 elements which is greater than 0. So, there are 19 numbers which are greater than zero
১,২১৯.
The average of four consecutive even numbers is 27. Find the smallest of these numbers.
  1. ক
    24
  2. খ
    30
  3. গ
    20
  4. ঘ
    40
ব্যাখ্যা

Consider the consecutive even numbers as : x, (x + 2), (x + 4) and (x+ 6)
Average = Sum of Quantities/Number of Quantities
{x + (x + 2) + (x + 4) + (x + 6)}/4 = 27
⇒ (4x + 12)/4 = 27
⇒ x + 3 = 27
⇒ x = 27 - 3
⇒ x = 24.

Therefore,
Largest number = (x + 6) = (24 + 6) = 30
Smallest number = 24.
Hence, the answer is 24.

১,২২০.
√(0.01 + √0.0064)=?
  1. ক
    ক) 0.03
  2. খ
    খ) 0.42
  3. গ
    গ) 0.3
  4. ঘ
    ঘ) None of these
ব্যাখ্যা
√(0.01 + √0.0064)
= √(0.01 + 0.08)
= √(0.09)
= 0.3
১,২২১.
If x is an odd integer, which of the following is true?
  1. ক
    5x - 2 is even
  2. খ
    5x2 + 2 is odd
  3. গ
    5x3 + 3 is odd
  4. ঘ
    None of these
ব্যাখ্যা
Question: If x is an odd integer, which of the following is true?

Solution:
let, x = 1
putting x to each option we get,
5x - 2 = (5 × 1) - 2 = 3; here 3 is an odd integer. so, option 1 is not true.
5x2 + 2 = (5 × 1) + 2 = 7; here 7 is an odd integer. so, option 2 is true.
5x2 + 3 = (5 × 1) + 3 = 8; here 8 is an even integer. so, option 3 is not true.

so only option 2 is true.
১,২২২.
The cost of 9 mangoes and 5 apples is equal to the cost of 7 mangoes and 8 apples. Find the ratio between the cost of 1 mango and the cost of one apple.
  1. ক
    5 : 8
  2. খ
    5 : 4
  3. গ
    7 : 5
  4. ঘ
    3 : 2
ব্যাখ্যা
Question: The cost of 9 mangoes and 5 apples is equal to the cost of 7 mangoes and 8 apples. Find the ratio between the cost of 1 mango and the cost of one apple.

Solution:
ধরি,
১টি আমের দাম ক টাকা 
১টি আপেলের দাম খ টাকা 

শর্তমতে,
৯ক + ৫খ = ৭ক + ৮খ
⇒ ৯ক - ৭ক = ৮খ - ৫খ 
⇒ ২ক = ৩খ 
⇒ ক/খ = ৩/২
∴ ক : খ = ৩ : ২ 
১,২২৩.
If the sum of two numbers is 30 and the sum of their squares is 692, then the product of two numbers is -
  1. ক
    ক) 81
  2. খ
    খ) 144
  3. গ
    গ) 104
  4. ঘ
    ঘ) 125
ব্যাখ্যা
Question: If the sum of two numbers is 30 and the sum of their squares is 692, then the product of two numbers is -

Solutuion:
ধরি,
একটি সংখ্যা = x
অপরটি = y

প্রশ্নমতে,
x + y = 30 ........... (i)
x2 + y2 = 692 ........... (ii)

(ii) নং হতে পাই,
x2 + y2 = 692
⇒ (x + y)2 - 2xy = 692
⇒ (30)2 - 2xy = 692
⇒ 900 - 2xy = 692
⇒ 2xy = 208
∴ xy = 104
১,২২৪.
If p is a positive integer, what is the smallest possible value of p such that 1470 × p is a perfect square?
  1. ক
    4
  2. খ
    12
  3. গ
    15
  4. ঘ
    30
ব্যাখ্যা

Question: If p is a positive integer, what is the smallest possible value of p such that 1470 × p is a perfect square?

Solution:
আমরা জানি, একটি সংখ্যা পূর্ণবর্গ হতে হলে তার মৌলিক গুণনীয়কের ঘাতসমূহ সবই জোড় সংখ্যা হতে হবে।

1470 = 2 × 3 × 5 × 7 × 7
= 21 × 31 × 51 × 72

এখন, 1470 × p = 21 × 31 × 51 × 72 × p
এখানে, 2-এর ঘাত = 1 (বিজোড়), 3-এর ঘাত = 1 (বিজোড়), 5-এর ঘাত = 1 (বিজোড়), 7-এর ঘাত = 2 (জোড়)
পূর্ণবর্গ করতে হলে সব ঘাত জোড় হতে হবে। তাই p = 2 × 3 × 5 = 30 হলে,
1470 × 30 = 21 × 31 × 51 × 72 × (2 × 3 × 5)
= 22 × 32 × 52 × 72

যেহেতু সব মৌলিক উৎপাদকের ঘাত জোড়, তাই এটি একটি পূর্ণবর্গ সংখ্যা।

সুতরাং, p = 30 হলে, 1470p পূর্ণবর্গ সংখ্যা হয়।

১,২২৫.
What is the odd man out?
835, 734, 642, 751, 853, 981, 532
  1. ক
    835
  2. খ
    734
  3. গ
    642
  4. ঘ
    751
  5. ঙ
    853
ব্যাখ্যা
In each number except 751, the difference of third and first digit is the middle one.
১,২২৬.
When the positive integers x and y are divided by the positive integer z, they yield remainders 12 and 22, respectively. If (x + y) is divided by z, the remainder is 6. What is the value of z?
  1. ক
    12
  2. খ
    15
  3. গ
    24
  4. ঘ
    28
  5. ঙ
    None
ব্যাখ্যা

Question: When the positive integers x and y are divided by the positive integer z, they yield remainders 12 and 22, respectively. If (x + y) is divided by z, the remainder is 6. What is the value of z?

Solution: 
ধরি, x কে z দিয়ে ভাগ করলে ভাগশেষ 12
অর্থাৎ, x হলো z এর কোনো গুণিতক থেকে 12 বেশি।

একইভাবে,
y কে z দিয়ে ভাগ করলে ভাগশেষ 22
অর্থাৎ, y হলো z এর কোনো গুণিতক থেকে 22 বেশি।
তাই (x + y) এর ভাগশেষ আগে হবে = 12 + 22 = 34

কিন্তু দেওয়া আছে, (x + y) কে z দিয়ে ভাগ করলে ভাগশে 6।
অর্থাৎ,
⇒ 34 - z = 6
⇒ z = 34 - 6
∴ z = 28

১,২২৭.
Twice the difference between two numbers is equal to their sum. If one number is 15, find the other number.
  1. ক
    5
  2. খ
    10
  3. গ
    15
  4. ঘ
    20
ব্যাখ্যা
Question: Twice the difference between two numbers is equal to their sum. If one number is 15, find the other number.

Solution:
Let,
Other number be x

ATQ,
x + 15 = 2(15 - x)
⇒ x + 15 = 30 - 2x
⇒ x + 2x = 30 - 15
⇒ 3x = 15
∴ x = 5
১,২২৮.
What least number must be added to 1056, So that the sum is completely divisible by 23?
  1. ক
    ক) 2
  2. খ
    খ) 3
  3. গ
    গ) 18
  4. ঘ
    ঘ) 21
ব্যাখ্যা

1056 ÷ 23, quotient = 45, remainder = 21
Since the remainder is not zero.
So, 1056 is not exactly divisible by 23
So, we take the next multiple of 23. Next multiple = 46
So, 23 X 46 = 1058
So, 1058 is exactly divisible by 23.
1958 - 1056 = 2
Hence, 2 is the least number to be added to 1056.

১,২২৯.
The difference between a two-digit number and the number obtained by interchanging the two digits is 63. Which is the smaller of the two numbers?
  1. ক
    ক) 29
  2. খ
    খ) 70
  3. গ
    গ) 92
  4. ঘ
    ঘ) Can not be determined
ব্যাখ্যা

Let the ten's digit be x and unit's digit be y.
Then, (10x + y) - (10y + x) = 63
⇔ 9 (x - y) = 63
x - y = 7.
There are several numbers like this, e.g. 70-07, 81-18 and 92-29. Thus, the correct answer is - ঘ) Can not be determined

তবে, ঘ) Can not be determined এই অপশন না থাকলে 29 কে উত্তর হিসেবে নেয়া যেত।
১,২৩০.
Which of the following numbers is a divisor of (4915 - 1)?
  1. ক
    8
  2. খ
    14
  3. গ
    48
  4. ঘ
    50
ব্যাখ্যা
Question: Which of the following numbers is a divisor of (4915 - 1)?

Solution:
an - bn is divisible by (a + b) when n is an even positive integer.
Here, a & b should be prime number.

(4915 - 1)
⇒ {(72)15 - 1)
⇒ (730 - 1)
Here, 30 is a positive integer.
According to the concept, (730 - 1) is divisible by (7 + 1) i.e., 8.
∴ 8 is a divisor of (4915 - 1). 
১,২৩১.
Which is the smallest fraction?
  1. ক
    ক) 5/13
  2. খ
    খ) 18/36
  3. গ
    গ) 16/31
  4. ঘ
    ঘ) 4/12
ব্যাখ্যা
Question: Which is the smallest fraction?

Solution: 
5/13 = 0.385

18/36
= 1/2
= 0.5

16/31
= 0.516 

4/12
= 1/3
= 0.33

So, 4/12 is the smallest fraction.
১,২৩২.
If n/23 is 2 more than m/23, then n = ?
  1. ক
    m + 23
  2. খ
    m - 41
  3. গ
    m + 46
  4. ঘ
    m + 42
ব্যাখ্যা
Question: If n/23 is 2 more than m/23, then n = ?

Solution:
n/23 = (m/23) + 2
⇒ n/23 = (m + 46)/23
∴ n = m + 46
১,২৩৩.
Quantity A = (- 6)4 and Quantity B = (- 6)5
  1. ক
    ক) Quantity A is greater
  2. খ
    খ) Quantity B is greater
  3. গ
    গ) Two quantities are equal
  4. ঘ
    ঘ) The relationship indeterminate
  5. ঙ
    ঙ) None of these
ব্যাখ্যা
Question: Quantity A = (- 6)4 and Quantity B = (- 6)5

Solution:
Quantity A = (- 6)4
= 1296

Quantity B = (- 6)5
= - 7776

∴ Quantity A is greater.
১,২৩৪.
The smallest number added to 680621 to make the sum a perfect square is:
  1. ক
    4
  2. খ
    5
  3. গ
    6
  4. ঘ
    8
ব্যাখ্যা

Question: The smallest number added to 680621 to make the sum a perfect square is:

Solution:
এখানে
(825)2 = 825 × 825
= 680625

প্রদত্ত সংখ্যা = 680621
নির্ণেয় ক্ষুদ্রতম সংখ্যা = (680625 - 680621) = 4

১,২৩৫.
Three numbers are in the ratio 1 : 2 : 3 and their H.C.F 14. The smallest number is-
  1. ক
    ক) 14
  2. খ
    খ) 28
  3. গ
    গ) 42
  4. ঘ
    ঘ) 84
ব্যাখ্যা
Given that 
The ratio of three numbers 1 : 2 : 3
Let the numbers be x, 2x and 3x.
The HCF in x, 2x and 3x is x

Hence,
x = 14;
 
The smallest number is 14
১,২৩৬.
If 60 is divided into two parts in such a way that the sum of their reciprocals is 3/40, the difference between the two parts is -
  1. ক
    ক) 15
  2. খ
    খ) 18
  3. গ
    গ) 20
  4. ঘ
    ঘ) 22
ব্যাখ্যা
Question: If 60 is divided into two parts in such a way that the sum of their reciprocals is 3/40, the difference between the two parts is -

Solution: 
বড় সংখ্যাটি = x 
ছোট সংখ্যাটি = y 

প্রশ্নমতে 
x + y = 60 ...................(1)

আবার
(1/x) + (1/y) =3/40
⇒ (y + x)/xy = 3/40
⇒ 60/xy = 3/40
⇒ 3xy = 2400
⇒ xy = 800

আমরা জানি 
(x - y)2 = (x + y)2 - 4xy 
⇒ (x - y)2 = (60)2 - 4 × 800 
⇒ (x - y)2 = 3600 - 3200
⇒ (x - y)2 = 400
⇒ (x - y)2 =202
⇒ (x - y) = 20
১,২৩৭.
If the sum of two numbers is 22 and the sum of their squares is 404, then the product of two numbers is
  1. ক
    ক) 40
  2. খ
    খ) 44
  3. গ
    গ) 80
  4. ঘ
    ঘ) 88
ব্যাখ্যা
ধরি,
একটি সংখ্যা x অপরটি y
প্রশ্নমতে, x + y = 22 ........... (i)
x2 + y2 = 404 ........... (ii)
(ii) নং হতে পাই, x2 + y2 = 404
⇒ (x + y)2 - 2xy = 404
⇒ (22)2 - 2xy = 404
⇒ 484 - 2xy = 404
⇒ 2xy = 80
⇒ xy = 40
১,২৩৮.
2 + x√3 = 1/(2 + √3), x = ? 
  1. ক
    1
  2. খ
    -1
  3. গ
    -√3
  4. ঘ
    √3
ব্যাখ্যা
Question: 2 + x√3 = 1/(2 + √3), x = ? 

Solution: 
2 + x√3 = 1/(2 + √3) 
⇒ 2 + x√3 = (2 - √3)/(2 + √3)(2 - √3) 
⇒ 2 + x√3 = (2 - √3)/{22 - (√3)2}
⇒ 2 + x√3 = (2 - √3)/(4 - 3)
⇒ 2 + x√3 = 2 - √3
⇒  x√3 = - √3
∴ x = - 1
১,২৩৯.
(10)2 is how many times of (0.01)3?
  1. ক
    105
  2. খ
    106
  3. গ
    107
  4. ঘ
    108
ব্যাখ্যা
Question: (10)2 is how many times of (0.01)3?

Solution:
(10)2/(0.01)3
= (10)2/(1/100)3
= 102/(1/102)3
= 102/(1/106)
= 102 × 106
= 102 + 6
= 108
১,২৪০.
The sum of two numbers is 35. Their difference is 1/7 of their sum. Their LCM is -
  1. ক
    ক) 60
  2. খ
    খ) 80
  3. গ
    গ) 100
  4. ঘ
    ঘ) 120
ব্যাখ্যা
Let
the number be x and y where x > y
According to the question,
x + y = 35.............(1)
 x - y = (1/7​)(x + y)

⇒ x - y = 35/7​ = 5    [From equation (1)]
x - y = 5...........(2)

From(1) and (2)
x = 20; y = 15

Hence, LCM of 20 and 15 = 60
১,২৪১.
Which of the following terms does not describe the number 9?
  1. ক
    ক) Prime
  2. খ
    খ) Integer
  3. গ
    গ) Real Number
  4. ঘ
    ঘ) Whole Number
ব্যাখ্যা
9 is not a prime number but it is an integer, real and whole number.
১,২৪২.
Find the HCF of (3125-1) and (335-1).
  1. ক
    ক) 34 - 1
  2. খ
    খ) 35 - 1
  3. গ
    গ) 312 - 1
  4. ঘ
    ঘ) None of these
ব্যাখ্যা

The solution of this question is based on the rule,
The HCF of (am - 1) and (an - 1) is given by (aHCF of m, n - 1)
Thus for this question the answer is (35 - 1)
Since, 5 is the HCF of 35 and 125

১,২৪৩.
What number multiplied by 48 will give the same product as 173 multiplied by 240?
  1. ক
    ক) 545
  2. খ
    খ) 665
  3. গ
    গ) 865
  4. ঘ
    ঘ) 685
ব্যাখ্যা
Question: What number multiplied by 48 will give the same product as 173 multiplied by 240?

Solution:
মনে করি,
সংখ্যাটি = x

প্রশ্নমতে,
x × 48 = 173 × 240
বা, x = (173 × 240)/48
∴ x = 865

∴ সংখ্যাটি 865
১,২৪৪.
If (a-b) is 6 more than (c+d) and (a+b) is 3 less than (c-d),Than ( a-c) is -
  1. ক
    ক) .5
  2. খ
    খ) 1
  3. গ
    গ) 1.5
  4. ঘ
    ঘ) None of these
ব্যাখ্যা

As per statement (a - b) is 6 more than (c + d).
a - b = (c + d) + 6 ............ (1)
As per statement (a + b) is 3 less than (c - d)
a + b = (c - d) - 3 ........... (2)

Adding both equations Eq(1) and Eq(2)
(a - b) + (a + b) = (c + d) + 6 + (c - d) - 3
⇒ a - b + a + b = c + d + 6 + c - d - 3
⇒ 2a = 2c + 3
⇒ 2a - 2c = 3
⇒ 2 (a - c) = 3
⇒ a - c = 3/2
∴ a - c = 1.5

১,২৪৫.
Let m be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in m is:
  1. ক
    ক) 2
  2. খ
    খ) 3
  3. গ
    গ) 4
  4. ঘ
    ঘ) 7
ব্যাখ্যা
m = H.C.F. of (4665 - 1305), (6905 - 4665) and (6905 - 1305)
Therefore, H.C.F. of 3360, 2240 and 5600 is 1120.
m = 1120
Sum of digits in m = ( 1 + 1 + 2 + 0 ) = 4
-----------------------------------------------------------------
প্রশ্নে বলা হয়েছে যে, m এমন একটি বৃহত্তম সংখ্যা যা দ্বারা ৪৬৬৫, ১৩০৫ ও ৬৯০৫ কে ভাগ করলে, প্রত্যেক ক্ষেত্রে একই ভাগশেষ থাকবে। m এর অঙ্কগুলোর সমষ্টি কত?
এখানে m হচ্ছে (৪৬৬৫ - ১৩০৫), (৬৯০৫ - ৪৬৬৫) ও (৬৯০৫ - ১৩০৫) এর গসাগু = ১১২০
m = ১১২০
m এর অঙ্কগুলোর সমষ্টি = ১ + ১ + ২ + ০ = ৪
১,২৪৬.
When n is divided by 4, the remainder is 3. What is the remainder when 2n is divided by 4 ? 
  1. ক
    ক) 1
  2. খ
    খ) 2
  3. গ
    গ) 6
  4. ঘ
    ঘ) 4
ব্যাখ্যা
Let
n = 4k + 3
2n = 2(4k + 3)
     = 8k + 6
     = 4 × 2k + 4 × 1 + 2
      = 4(2k + 1) + 2
Thus, on dividing 2n by 4, we get 2 as remainder .
১,২৪৭.
The difference between the average of all prime numbers between 30 and 60 and the average of all prime numbers between 15 and 30 is-
  1. ক
    21.4
  2. খ
    22.4
  3. গ
    18.7
  4. ঘ
    23.8
ব্যাখ্যা

Question: The difference between the average of all prime numbers between 30 and 60 and the average of all prime numbers between 15 and 30 is-

Solution:
The prime numbers between 30 and 60 are 31, 37, 41, 43, 47, 53 and 59.

∴ The average of all prime numbers between 30 and 60 is,
= (31 + 37 + 41 + 43 + 47 + 53 + 59)/7
= 311/7
= 44.43

And,
The prime numbers between 15 and 30 are 17, 19, 23 and 29.

∴ The average of all prime numbers between 15 and 30 is,
= (17 + 19 + 23 + 29)/4
= 88/4
= 22

∴ Required difference = (44.43) - 22 = 22.4

১,২৪৮.
The difference between a positive proper fraction and its reciprocal is 9/20. The fraction is-
  1. ক
    3/5
  2. খ
    3/10
  3. গ
    4/5
  4. ঘ
    4/3
ব্যাখ্যা
Question: The difference between a positive proper fraction and its reciprocal is 9/20. The fraction is-

Solution:
Let the required fraction be x.
Then,
1/x - x =9/20
⇒ (1 - x2)/x = 9/20
⇒ 20 - 20x2 = 9x
⇒ 20x2 + 9x - 20 = 0
⇒ 20x2 + 25x - 16x - 20 = 0
⇒ 5x(4x + 5) - 4(4x + 5) = 0
⇒ (4x + 5)(5x - 4) = 0
4x + 5 = 0   or  5x - 4 = 0
⇒ x = - 5/4  or x = 4/5  [x = - 5/4 not acceptable]
∴ x = 4/5
১,২৪৯.
In a library system with 6 branches, 60 workers are employed. If no library has fewer than 7 workers and no more than 11, what is the minimum number of workers in any two of the branches?
  1. ক
    ক) 10
  2. খ
    খ) 14
  3. গ
    গ) 19
  4. ঘ
    ঘ) 16
ব্যাখ্যা
Question: In a library system with 6 branches, 60 workers are employed. If no library has fewer than 7 workers and no more than 11, what is the minimum number of workers in any two of the branches?

Solution:
In 4 branches maximum 11 × 4 = 44 workers can be employed.
Rest 2 branches minimum 60 - 44 = 16 workers can be employed.
১,২৫০.
The largest 5 digit number exactly divisible by 91 is-
  1. ক
    99918
  2. খ
    99921
  3. গ
    99981
  4. ঘ
    99971
ব্যাখ্যা
Question: The largest 5 digit number exactly divisible by 91 is-

Solution:
Largest 5 digit number = 99999

99999 ÷ 91 (Quotient = 1098; Remainder = 81)

∴ Required number =(99999 - 81)
= 99918
১,২৫১.
Which of the following numbers is a prime number?
  1. ক
    167
  2. খ
    213
  3. গ
    352
  4. ঘ
    437
  5. ঙ
    None of these
ব্যাখ্যা
Question: Which of the following numbers is a prime number?

Solution:
167 is not divisible by any prime number

213 is divisible by 3

352 is divisible by 2 and 11

437 is divisible by 19

∴ 167 is the required prime number as it is not divisible by any prime number.
১,২৫২.
3 x 0.3 x 0.03 x 0.003 x 30 = ?
  1. ক
    0.0000243
  2. খ
    0.000243
  3. গ
    0.00243
  4. ঘ
    0.0243
  5. ঙ
    None of the above
ব্যাখ্যা
3 x 3 x 3 x 3 x 30 = 2430. Sum of decimal places = 6
Therefore, 3 x 0.3 x 0.03 x 0.003 x 30 = 0.002430 = 0.00243
১,২৫৩.
Which of the following is the largest?
  1. ক
    227
  2. খ
    1253
  3. গ
    518
  4. ঘ
    44.1024
  5. ঙ
    279
ব্যাখ্যা
Question:  Which of the following is the largest?

Solution:
Let's simplify each option to comparable base powers:
A) 227 already in base 2
B) 1253 = (53)3 = 59
C) 518 already in base 5
D) 44 × 1024=(22)4× 210 = 28× 210 = 218
E) 279 = (33)9 = 327

Now observe:
518 is clearly much larger than 59
227 is larger than 218

But base 3 raised to 27 (i.e., 327) grows faster than all

Since 327 increases faster than any of the others, option E is the largest.
১,২৫৪.
The greatest number that exactly divides 105, 1001 and 2436 is
  1. ক
    ক) 3
  2. খ
    খ) 7
  3. গ
    গ) 11
  4. ঘ
    ঘ) 21
ব্যাখ্যা
বৃহত্তম সংখ্যাটি হবে ১০৫, ১০০১ এবং ২৪৩৬ এর গ.সা.গু 

এখানে,
১০৫ = ৩ × ৫ × ৭ 
১০০১ = ৭ × ১১ × ১৩ 
২৪৩৬ =  ২ × ২ × ৩ × ৭ × ২৯ 

নির্ণেয় গ.সা.গু = ৭ 

অতএব, 
বৃহত্তম সংখ্যাটি ৭
১,২৫৫.
The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. The square of their sum is:
  1. ক
    420
  2. খ
    320
  3. গ
    360
  4. ঘ
    400
ব্যাখ্যা
Question: The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. The square of their sum is:
 
Solution:
Let the numbers be a, b and c

Then,
a2 + b2 + c2 = 138
(ab + bc + ca) = 131
 
Now,
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
⇒ (a + b + c)2 = 138 + 2 × 131
∴ (a + b + c)2 = 400
১,২৫৬.
The product of two numbers is 168 and the sum of their square is 289. The sum of the numbers is:
  1. ক
    21
  2. খ
    22
  3. গ
    24
  4. ঘ
    25
ব্যাখ্যা
Question: The product of two numbers is 168 and the sum of their square is 289. The sum of the numbers is:

Solution:
Let
the numbers be x and y

Then,
xy = 168
and x2 + y2 = 289

∴ (x + y)2 = x2 + y2 + 2xy
= 289 + (2 × 168)
= 289 + 336
= 625

∴ x + y = √625 = 25
১,২৫৭.
25/? = ?/81. What is the missing number?
  1. ক
    ক) 2025
  2. খ
    খ) 45
  3. গ
    গ) 53
  4. ঘ
    ঘ) 49
  5. ঙ
    ঙ) 65
ব্যাখ্যা
Question: 25/? = ?/81. What is the missing number?

Solution: 
25/? = ?/81
⇒ ?2 = 25 × 81
⇒ ? = 5 × 9
∴ ? = 45
১,২৫৮.
The number of prime factors in the expression 610 × 516 × 1318 is equal to-
  1. ক
    54
  2. খ
    64
  3. গ
    44
  4. ঘ
    62
  5. ঙ
    None of these
ব্যাখ্যা
Question: The number of prime factors in the expression 610 × 516 × 1318 is equal to-

Solution:
610 × 516 × 1318
= (2 × 3)10 × 516 × 1318
= 210 × 310 × 516 × 1318

∴ Number of prime factors in the given expression = (10 + 10 + 16 + 18) = 54
১,২৫৯.
2/0 is?
  1. ক
    1
  2. খ
    0
  3. গ
    2
  4. ঘ
    Undefiend
ব্যাখ্যা
Question: 2/0 is?

Solution: 
2/0 is Undefiend

কোন সংখ্যাকে শূন্য দ্বারা ভাগ করলে তা অসংজ্ঞায়িত হয়।
১,২৬০.
When 4 is added to a number 1/2 of a number, it results in 14. What is the number?
  1. ক
    ক) 20
  2. খ
    খ) 21
  3. গ
    গ) 27
  4. ঘ
    ঘ) 35
  5. ঙ
    ঙ) None of the above
ব্যাখ্যা

Let,
The number is x,
therefore, (1/2) x + 4 = 14,
so, x = 20

১,২৬১.
A student multiplied 765 by a certain number and got 448835 as their answer. If in the answer both 8 is wrong, but the other digits are correct, then what will be the correct?
  1. ক
    446435
  2. খ
    445935
  3. গ
    444635
  4. ঘ
    442935
  5. ঙ
    None of the above
ব্যাখ্যা
Question: A student multiplied 765 by a certain number and got 448835 as their answer. If in the answer both 8 is wrong, but the other digits are correct, then what will be the correct?

Solution:
The answer is divisible by 765.
So we can use the hit and trial method to find out the number divisible by 765 from the given choices.
446435/765 gives a remainder not equal to 0
445935/765 gives a remainder not equal to 0
444635/765 gives a remainder not equal to 0
But 442935/765 gives 0 as a remainder (equals 579). Hence this is the answer.
১,২৬২.
What is the greatest common factor of 24 and 64?
  1. ক
    ক) 4
  2. খ
    খ) 8
  3. গ
    গ) 12
  4. ঘ
    ঘ) 36
ব্যাখ্যা
24 = 2 × 2 × 2 × 3
64 = 2 × 2 × 2 × 2 × 2 × 2
So, greatest common factor of 24 and 64 is 2 × 2 × 2 = 8
১,২৬৩.
The sum of five consecutive numbers is 85. What is the difference between thrice the smallest number and twice the largest number?
  1. ক
    6
  2. খ
    7
  3. গ
    8
  4. ঘ
    9
ব্যাখ্যা
Question: The sum of five consecutive numbers is 85. What is the difference between thrice the smallest number and twice the largest number?

Solution:
Let the smallest number be x
Then the five consecutive numbers are: x, x + 1, x + 2, x + 3, x + 4
ATQ,
x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 85
⇒ 5x + 10 = 85
⇒ 5x = 75
⇒ x = 15

∴ Difference between thrice the smallest number and twice the largest number
= (x × 3) - {(x + 4) × 2}
= (15 × 3) - {(15 + 4) × 2}
= 7
১,২৬৪.
Six consercutive whole numbers are given. The sum of last three numbers is 36. What is the sum of the first three numbers?
  1. ক
    32
  2. খ
    30
  3. গ
    28
  4. ঘ
    27
ব্যাখ্যা
Question: Six consercutive whole numbers are given. The sum of last three numbers is 36. What is the sum of the first three numbers?

Solution:
Let,
the numbers be x - 2 , x - 1 , x, x + 1 , x + 2, x + 3

ATQ,
x + 1 + x + 2 + x + 3 = 36
⇒ 3x + 6 = 36
⇒ 3x = 36 - 6
⇒ 3x = 30
∴ x = 10

the sum of the last three numbers is = (x - 2) + (x - 1) + x 
= (10 - 2) + (10 - 1) + 10
= 8 + 9 + 10
= 27
১,২৬৫.
Through what angle does the minute hand of a clock turn in 5 minutes?
  1. ক
    ক) 30∘
  2. খ
    খ) 32∘
  3. গ
    গ) 35∘
  4. ঘ
    ঘ) 36∘
ব্যাখ্যা

Angle traced by the minute hand in 5 minutes. =(360/60×5)∘ = 30∘

১,২৬৬.
One third of the faculty members of a department are female, Sixteen of the male teachers' are unmarried, while 60% of them are married. The total number of faculty members in the department is
  1. ক
    ক) 72
  2. খ
    খ) 60
  3. গ
    গ) 30
  4. ঘ
    ঘ) 90
ব্যাখ্যা
পুরুষ হলো = (1 - 1/3) অংশ
                 =(3 - 1)/3
                 = 2/3 অংশ 

অবিবাহিত পুরুষ হলো = 100% - 60% = 40%

ধরি 
পুরুষ = x জন 

এখন 
x এর 40% = 16
40x/100 = 16 
2x/5 = 16
x = 16 × (5/2)
x = 40 

আবার ,
2/3 অংশ  = 40
1 বা সম্পূর্ণ অংশ =  (40× 3)/2 = 60 জন
১,২৬৭.
A student first reduced a number by 20 percent and increased it again by 20 percent. If the difference between the two new numbers was 8, then what is the original number?
  1. ক
    30
  2. খ
    40
  3. গ
    50
  4. ঘ
    80
ব্যাখ্যা
Question: A student first reduced a number by 20 percent and increased it again by 20 percent. If the difference between the two new numbers was 8, then what is the original number?

Solution:
Let, the original number = 100

First,
20% reduced number = 100 - 20 =80
Again,
20% increased number = 80 + 80 এর 20%
= 80 + 80 (20/100) = 96

So, the difference = (96 - 80) = 16

16 difference of = 100
1 difference of = 100/16
8 difference of = (8 100)/16
= 50 Taka
∴ The original number = 50
১,২৬৮.
If n is an integer between 10 and 70, then any of the following could be n + 7 except-
  1. ক
    78
  2. খ
    70
  3. গ
    57
  4. ঘ
    46
ব্যাখ্যা
Question: If n is an integer between 10 and 70, then any of the following could be n + 7 except-

Solution: 
maximum value of n + 7 = 69 + 7 = 76 
78 > 76
So, the correct answer is A
১,২৬৯.
If two-third of three-fourth of a number is 34, find the 20% of that number?
  1. ক
    13
  2. খ
    13.6
  3. গ
    14
  4. ঘ
    14.6
ব্যাখ্যা
Question: If two-third of three-fourth of a number is 34, find the 20% of that number?

Solution:
Let, the number be x.

ATQ,
(2/3) × (3/4) × x = 34
⇒ (1/2) × x = 34
∴ x = 68

20% of 68 = (20 × 68)/100 = 13.6
১,২৭০.
The 180 students in a group are to be seated in rows so that there is an equal number of students in each row. Each of the following could be the number of rows EXCEPT
  1. ক
    4
  2. খ
    20
  3. গ
    30
  4. ঘ
    40
  5. ঙ
    90
ব্যাখ্যা
Question: The 180 students in a group are to be seated in rows so that there is an equal number of students in each row. Each of the following could be the number of rows EXCEPT.

Solution:
Obviously the number of rows must be a factor of 180.
180/4=45
180/20=9
180/30=6
180/40=4.50
180/90 = 2

The only option which is not a factor of 180 is 40
১,২৭১.
If p is an even integer and q is an odd integer, which of the following must be an odd integer?
  1. ক
    ক) p/q
  2. খ
    খ) 2p + q
  3. গ
    গ) pq
  4. ঘ
    ঘ) 2 (p + q)
  5. ঙ
    ঙ) 3p/q
ব্যাখ্যা
Question: If p is an even integer and q is an odd integer, which of the following must be an odd integer?

Solution: 
2 × even integer = even integer
So, 2p is an even integer

even integer + odd integer = odd integer

So,
2p + q must be an odd integer.
১,২৭২.
The average of first 101 natural numbers is:
  1. ক
    51
  2. খ
    5151
  3. গ
    5252
  4. ঘ
    52
ব্যাখ্যা
The average of first 101 natural numbers
= (101 + 1)/2
= 51
১,২৭৩.
When a number is divided by 13, the remainder is 11. When the same number is divided by 17, then the remainder is 9. What is the number?
  1. ক
    ক) 339
  2. খ
    খ) 349
  3. গ
    গ) 359
  4. ঘ
    ঘ) 369
ব্যাখ্যা

13p + 11 and x = 17q + 9
∵ 13p + 11 = 17q + 9
17q - 13p = 2
q = (2 + 13p)/17
∵ The least value of p for which q = (2 + 13p)/17 is a whole number p = 26
x = (13 x 26 + 11)
= (338 + 11)
= 349

১,২৭৪.
Which of the following is divisible by 2 and 7?
  1. ক
    ক) 365
  2. খ
    খ) 362
  3. গ
    গ) 361
  4. ঘ
    ঘ) None
ব্যাখ্যা
Question: Which of the following is divisible by 2 and 7?

Solution: 
২, ৭ এর ল. সা. গু = ১৪ 

৩৬৫, ৩৬২, ৩৬১ ; ১৪ দ্বারা বিভাজ্য নয়। 
১,২৭৫.
What smallest number should be added to 4456 so that the sum is completely divisible by 6?
  1. ক
    2
  2. খ
    3
  3. গ
    4
  4. ঘ
    None of these
ব্যাখ্যা
Question: What smallest number should be added to 4456 so that the sum is completely divisible by 6?

Solution: 
4456 divided by 6 leaves a remainder of 4. Therefore, we need to add 2 to 4456 to make the sum divisible by 6.

4456 + 2 = 4458, and when divided by 6, this results in a remainder of 0.
১,২৭৬.
The number, whose square is equal to the difference of the squares of 75.15 and 60.12, is = ?
  1. ক
    ক) 56.39
  2. খ
    খ) 88.06
  3. গ
    গ) 45.09
  4. ঘ
    ঘ) 47.06
ব্যাখ্যা

Let the number be = x
According to question,
x2=(75.15)2−(60.12)2
⇒x2=(75.15+60.12)(75.15−60.12)
⇒x2=135.27×15.03
⇒x2=2033.1081
⇒x=45.09

১,২৭৭.
√(0.25/0.0009) × √(0.09/0.36) is equal to?
  1. ক
    ক) 5/6
  2. খ
    খ) 7(1/6)
  3. গ
    গ) 7(1/3)
  4. ঘ
    ঘ) 2(5/3)
  5. ঙ
    ঙ) 8(1/3)
ব্যাখ্যা

According to question,
√(0.25/0.0009) × √(0.09/0.36)
⇒ √((25/9)×100) × √(9/36)
⇒ ((5×10)/3) × (3/6)
⇒ 25/3
⇒ 8(1/3)

১,২৭৮.
Find the value of 6(-3)(1/3)(-0.25)
  1. ক
    ক) 6
  2. খ
    খ) 4.5
  3. গ
    গ) 1.5
  4. ঘ
    ঘ) -0.5
ব্যাখ্যা

6(-3)(1/3)(-0.25)
= 6×0.25
= 1.5

১,২৭৯.
The difference between two integers is 4. If their product is 221, then the sum of the two numbers is?
  1. ক
    30
  2. খ
    18
  3. গ
    26
  4. ঘ
    22
ব্যাখ্যা

Question: The difference between two integers is 4. If their product is 221, then the sum of the two numbers is?

Solution:
Let the two integers be x and y
The difference between the two integers is 4,
∴ x - y = 4 and their product is, xy = 221

We know,
⇒ (x + y)2 = (x - y)2 + 4xy
⇒ (x + y)2 = 42 + (4 × 221)
⇒ (x + y)2 = 16 + 884
⇒ (x + y)2 = 900
⇒ x + y = √900
∴ x + y = 30

∴ The sum of the two numbers is 30

১,২৮০.
If the sum of two numbers is 15 and their difference is 5. Find the two numbers.
  1. ক
    10, 5
  2. খ
    7, 8
  3. গ
    9, 6
  4. ঘ
    11, 4
ব্যাখ্যা
Question: If the sum of two numbers is 15 and their difference is 5. Find the two numbers.

Solution:
Let the two numbers be x and y. Then,
x + y = 15 .......(1)
x − y = 5 .......(2)

Adding equation (1) and (2), we get,
2x = 20
∴ x = 10

Thus, y = 5
Hence, the required numbers are 10 and 5.
১,২৮১.
What is the greatest number that divides 84, 144 or 18 without any remainder?
  1. ক
    ক) 6
  2. খ
    খ) 12
  3. গ
    গ) 18
  4. ঘ
    ঘ) 24
ব্যাখ্যা

HCF of the given numbers will be the greatest number which can divide 48, 84 and 144
18 = 2 × 3 × 3
84 = 2 × 2 × 3 × 7
144 = 2 × 2 × 2 × 2 × 3 × 3
∴ HCF = 2 × 3 = 6
Hence 6 is the greatest number which divides 18, 84 and 144 without leaving any remainder

১,২৮২.
A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 4, he could buy 10 books more. How much money does he has?
  1. ক
    ক) Tk. 1000
  2. খ
    খ) Tk. 1200
  3. গ
    গ) Tk. 1500
  4. ঘ
    ঘ) Tk. 2000
ব্যাখ্যা

Question: A shopkeeper has sufficient money to buy 50 books. On reduction in the price of each book by Tk. 4, he could buy 10 books more. How much money does he has?

Solution:
১টি বইয়ে দাম কমে ৪ টাকা
∴ ৫০টি বইয়ে দাম কমে (৫০ × ৪) টাকা 
= ২০০ টাকা 

সে মোট বই কিনে (৫০ + ১০) টি 
= ৬০টি

১০টি বইয়ের দাম ২০০ টাকা 
∴ ৬০টি বইয়ের দাম (২০০ × ৬০)/১০ টাকা 
= ১২০০ টাকা 

∴ তার কাছে ১২০০ টাকা আছে।

১,২৮৩.
If p is an even integer, which of the following must be an even integer? 
  1. ক
    p2 - p
  2. খ
    p + 2
  3. গ
    3p3
  4. ঘ
    All of the above
ব্যাখ্যা

Question: If p is an even integer, which of the following must be an even integer?

Solution:
ধরি,
p = 2

ক) p2 - p = 22 - 2 = 4 - 2 = 2 [যা জোড়]
খ) 3n3 = 3 × 23 = 24 [যা জোড়]
গ) p + 2 = 2 + 2 = 4 [যা জোড়]
∴ সঠিক উত্তর হচ্ছে ঘ) All of the above

১,২৮৪.
What is the least number of people who can be arranged in certain places of 12, 15, 18 and 20 people and also in form of a solid square?
  1. ক
    180
  2. খ
    450
  3. গ
    900
  4. ঘ
    1800
  5. ঙ
    2700
ব্যাখ্যা
In this type of question, We need to find out the L.C.M. of the given numbers.
L.C.M. of 12, 15, 18 and 20 = 180 = 3×3×2×2×5
L.C.M. must be a perfect square. To make the L.C.M. a perfect square, We have to multiply it by 5,
The required number of soldiers = 3×3×2×2×5×5 = 302 = 900
১,২৮৫.
In doing a question of division with zero remainder, a candidate took 12 divisor instead of 15. The quotient obtained by him was 35. The correct quotient is-
  1. ক
    17
  2. খ
    20
  3. গ
    24
  4. ঘ
    28
ব্যাখ্যা
Question: In doing a question of division with zero remainder, a candidate took 12 divisor instead of 15. The quotient obtained by him was 35. The correct quotient is-

Solution:
Divisor taken = 12
Quotient obtained = 35,
Remainder = 0
∴ Dividend = (12 × 35) = 420

Now, Dividend = 420,
Divisor = 15
Remainder = 0
∴ Quotient = 420/15
= 28
১,২৮৬.
If a and b are odd numbers. Which number is even?
  1. ক
    ab
  2. খ
    a + 2b + 2
  3. গ
    a + b + 1
  4. ঘ
    2a + 4b
ব্যাখ্যা
Question: If a and b are odd numbers. Which number is even?

Solution:
Let
a = 1
b = 3

ab = 1 × 3 = 3, which is odd.

a + 2b + 2 = 1 + 2 × 3 + 2 = 1 + 6 + 2 = 9, which is odd.

a + b + 1 = 1 + 3 + 1 = 5, which is odd.

2a + 4b = 2 × 1 + 4 × 3 = 2 + 12 = 14, which is even.
১,২৮৭.
Which of the following numbers is divisible by 3?
  1. ক
    177
  2. খ
    200
  3. গ
    331
  4. ঘ
    457
  5. ঙ
    None of these
ব্যাখ্যা
প্রশ্ন: Which of the following numbers is divisible by 3?

সমাধান:
ক. 177/3 = 59; যা 3 দ্ধারা বিভাজ্য।

খ. 200/3 = 66.67; যা 3 দ্ধারা বিভাজ্য নয়।

গ. 331/3 = 110.33; যা 3 দ্ধারা বিভাজ্য নয়।

ঘ. 457/3 = 152.33; যা 3 দ্ধারা বিভাজ্য নয়।

বিকল্প সমাধান:
কোনো সংখ্যার অঙ্কগুলোর যোগফল ৩ দ্বারা বিভাজ্য হলে, ঐ সংখ্যাটি ৩ দ্বারা বিভাজ্য হবে। 
ক. 1 + 7 + 7 = 15; যা 3 দ্ধারা বিভাজ্য।
খ. 2 + 0 + 0 = 2; যা 3 দ্ধারা বিভাজ্য নয়।
গ. 3 + 3 + 1 = 7; যা 3 দ্ধারা বিভাজ্য নয়।
ঘ. 4 + 5 + 7 = 16; যা 3 দ্ধারা বিভাজ্য নয়।
১,২৮৮.
Find the missing number
4, ?, 144, 400, 900, 1764 
  1. ক
    100
  2. খ
    49
  3. গ
    36
  4. ঘ
    25
ব্যাখ্যা
Question: Find the missing number
4, ?, 144, 400, 900, 1764 

Solution: 
1764 = 422
900 = (42 - 12)2 = 302
400 = (30 - 10)2 = 202
144 = (20 - 8)2 = 122 
? = (12 - 6)2 = 62 = 36
4 = (6 - 4)2 = 22

১ম বার 12 বিয়োগ দেয়া হয়েছে, (42 - 12),
তারপর 2 কমিয়ে 10 বিয়োগ দেয়া হয়েছে (30 - 10),
অনুরূপ প্রক্রিয়ায় সমাধান করে উত্তর হবে 36। 
১,২৮৯.
The L.C.M and ratio of four numbers are 630 and 2:3 and their 2:3:5:7 respectively. The difference between the greatest and least number is:
  1. ক
    ক) 6
  2. খ
    খ) 14
  3. গ
    গ) 15
  4. ঘ
    ঘ) 21
ব্যাখ্যা

Let the numbers be 2x, 3x, 5x and 7x respectively.
Then, their L.C.M = (2 × 3 × 5 × 7)x = 210x
[∵ 2, 3, 5, 7 are prime numbers ]
So, 20x = 630
or x = 3
∵ The numbers are 6, 9, 15 and 21.
Required difference = 21 - 6 = 15.
Answer : 15

১,২৯০.
What is the sum of the reciprocals of the values of zeroes of the polynomial 6x2 + 3x2 - 5x + 1?
  1. ক
    ক) 3
  2. খ
    খ) 4
  3. গ
    গ) 5
  4. ঘ
    ঘ) 6
ব্যাখ্যা
Question: What is the sum of the reciprocals of the values of zeroes of the polynomial 6x2 + 3x2 - 5x + 1?

Solution: 
6x2 + 3x2 - 5x + 1
⇒ 9x2 - 5x + 1
Let α and β are two roots of the equations

As we know,
Sum of roots (α + β) = 5/9
Product of roots (αβ) = 1/9
According to the question :
⇒ 1/α + 1/β = (α + β)/αβ
                     = (5/9)/(1/9)
                     = (5/9) × (9/1)
                     = 5
১,২৯১.
Which greatest number will divide 1304 and 1869 leaving remainders 8 and 9 respectively?
  1. ক
    ক) 12
  2. খ
    খ) 14
  3. গ
    গ) 16
  4. ঘ
    ঘ) 18
ব্যাখ্যা
Question: Which greatest number will divide 1304 and 1869 leaving remainders 8 and 9 respectively?

Solution:
The required number is = HCF of (1304 - 8) and (1869 - 9)
=  HCF of 1296 and 1860
= 12
১,২৯২.
The difference between the two numbers is 20% of the larger number. If the smaller number is 12, find the larger number.
  1. ক
    25
  2. খ
    20
  3. গ
    15
  4. ঘ
    10
ব্যাখ্যা
Question: The difference between the two numbers is 20% of the larger number. If the smaller number is 12, find the larger number.

Solution:
Let the number be x
ATQ,
x - 12 = 20% of x
⇒ x - 12 = x/5
⇒ x - x/5 = 12
⇒ 4x/5 = 12
⇒ x = (12 × 5)/4 
∴ x = 15
১,২৯৩.
The average of A and B is 45 and the sum of B & C is 78. What is the value of A - C?
  1. ক
    16
  2. খ
    12
  3. গ
    10
  4. ঘ
    22
ব্যাখ্যা

Question: The average of A and B is 45 and the sum of B & C is 78. What is the value of A - C?

Solution:
Given that,
Average of A and B = 45
Sum of B and C = 78

Now,
Average of A and B = 40, so-
⇒ (A + B)/2 = 45
∴ A + B = 90 ........(1)
And, B + C = 78 .........(2)

Subtract (2) from (1) than we get,
⇒ A + B - (B + C) = 90 - 78
⇒ A + B - B - C = 12
⇒ A - C = 12

So the value of A - C is 12.

১,২৯৪.
When an integer m is divided by 6, the remainder is 4. What is the remainder when 7m is divided by 3?
  1. ক
    3
  2. খ
    0
  3. গ
    2
  4. ঘ
    1
ব্যাখ্যা
Question: When an integer m is divided by 6, the remainder is 4. What is the remainder when 7m is divided by 3?

Solution:
m = 6 × quotient + remainder
or, m = 6n + 4    [Let, quotient = n]
or, 7m = 7 × 6n + 7 × 4
or, 7m = 7 × 6n + 28

since remainder can not be greater than or equal to divisor
= 6 ×(7n) + 27 + 1
= 3(14n + 9) + 1

if this number is divided by 3 the remainder will be 1

∴ Remainder is 1
১,২৯৫.
Find the remainder when 711 + 7111 + 71111 is divided by 8.
  1. ক
    1
  2. খ
    2
  3. গ
    3
  4. ঘ
    5
  5. ঙ
    7
ব্যাখ্যা

Question: Find the remainder when 711 + 7111 + 71111 is divided by 8.

Solution:
When, 7 is divided by 8 then remainder = 7
When, 72 is divided by 8 then remainder = 1
When, 73 is divided by 8 then remainder = 7
When, 74 is divided by 8 then remainder = 1
Odd exponents give remainder = 7
Even exponents give remainder = 1

In 1st term, exponent is 11, which is odd so remainder = 7
In 2nd term, exponent is 111, which is also odd so remainder = 7
In 3rd term, exponent is 1111, which is odd so remainder = 7
So, (711 + 7111 + 71111) mod 8 = (7 + 7 + 7) mod 8 = 21 mod 8 = 5

১,২৯৬.
Find the remainder when 496 is divided by 8.
  1. ক
    ক) 0
  2. খ
    খ) 1
  3. গ
    গ) 2
  4. ঘ
    ঘ) 3
ব্যাখ্যা
Question: Find the remainder when 496 is divided by 8.

Solution: 
496/8
= (42 × 494)/8
= (16 × 494)/8
= 2 × 494

যেহেতু 496, 8 দ্বারা নিঃশেষে বিভাজ্য। অতএব, ভাগশেষ শূন্য হবে। 
১,২৯৭.
0.004 × ? = 0.000016
  1. ক
    ক) 0.04
  2. খ
    খ) 0.004
  3. গ
    গ) 0.4
  4. ঘ
    ঘ) 0.0004
ব্যাখ্যা
0.004 × a = 0.000016
a = 0.000016/0.004
a = 0.004
১,২৯৮.
The difference between the greatest and the least four-digit numbers that begins with 3 and ends with 5 is -
  1. ক
    ক) 900
  2. খ
    খ) 990
  3. গ
    গ) 999
  4. ঘ
    ঘ) 909
ব্যাখ্যা
Question: The difference between the greatest and the least four-digit numbers that begins with 3 and ends with 5 is -

Solution:
The greatest four-digit number that begins with 3 and ends with 5 = 3995
The least four-digit number that begins with 3 and ends with 5 = 3005

∴ Required difference = 3995 - 3005 = 990
১,২৯৯.
What will be the least number which when tripled will be exactly divisible by 8, 12, 15?
  1. ক
    30
  2. খ
    45
  3. গ
    40
  4. ঘ
    32
ব্যাখ্যা

Question: What will be the least number which when tripled will be exactly divisible by 8, 12, 15?

Solution: 
Prime factorization of, 
8 = 2 × 2 × 2
12 = 2 × 2 × 3
15 = 3 × 5

LCM = 2 × 2 × 2 × 3 × 5 = 120
So the smallest, 3n = 120
⇒ n = 120/3 
∴ n = 40

So the least number which when tripled will be exactly divisible by 8, 12, and 15 is 40.

১,৩০০.
The product of two consecutive even numbers is 168. What are the numbers?
  1. ক
    12 and 14
  2. খ
    14 and 16
  3. গ
    16 and 18
  4. ঘ
    22 and 24
ব্যাখ্যা
Question: The product of two consecutive even numbers is 168. What are the numbers?

Solution:
Let the smaller number be x, 
so the next even number is = x + 2

According to the question,
 x(x + 2) = 168
⇒ x2 + 2x = 168
⇒ x2 + 2x -168 = 0 
⇒  x2 + 14x - 12x - 168 = 0
⇒ x(x + 14) - 12(x + 14) = 0 
⇒ (x + 14)(x - 12) = 0 

⇒  x = - 14 and x = 12

If x = 12,
then the next even number =
12 + 2 = 14
The two numbers are: 12 and 14.

Again, If x = - 14, then the next even number = - 14 + 2 = - 12 
 So, the numbers are 12 and 14 or
- 14 and - 12