Number System, Problems on Number

PrepBank · পাতা ১৫ / ১৮ · ১,৪০১–১,৫০০ / ১,৭৩৬

১,৪০১.

If one-third of one-fourth of a number is 25, then three-tenths of that number is:

ক
60
খ
55
গ
45
ঘ
90

ব্যাখ্যা

Question: If one-third of one-fourth of a number is 25, then three-tenths of that number is:

Solution:
Let the number be 'p'

Now,
(1/3) × (1/4) × p = 25
⇒ p/12 = 25
⇒ p = 12 × 25
∴ p = 300

∴ Three-tenths of that number will be = (3/10) × p
= (3/10) × 300
= 3 × 30
= 90

১,৪০২.

Find the least number when divided by 20, 25, 35, and 40 leaves remainders 15, 20, 30 and 35 respectively-

ক
ক) 1395
খ
খ) 1400
গ
গ) 1405
ঘ
ঘ) 1410

ব্যাখ্যা

Question: Find the least number when divided by 20, 25, 35, and 40 leaves remainders 15, 20, 30 and 35 respectively-

Solution:
Here
20 − 15 = 5
25 − 20 = 5
35 − 30 = 5
40 − 35= 5

Required number (L.C.M of 20, 25, 35, and 40) - 5

L.C.M of 20, 25, 35, and 40 = 1400

Required number= 1400 - 5 = 1395

১,৪০৩.

If n is a whole number greater then 1, then n² (n² - 1) is always divisible by -

ক
16
খ
10
গ
8
ঘ
12

ব্যাখ্যা

Question: If n is a whole number greater then 1, then n² (n² - 1) is always divisible by -

Solution:
Given that,
n²(n² - 1)
= 2²(2² - 1) ; [put n = 2]
= 4(4 - 1)
= 4 × 3
= 12
Check the option it is divisible by 12.
Take n = 3
= 3³(3² - 1)
= 9(9 - 1)
= 9 × 8
= 72
It is divisible by 12

১,৪০৪.

What is the least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder?

ক
ক) 1683
খ
খ) 1108
গ
গ) 2007
ঘ
ঘ) 3363

ব্যাখ্যা

LCM of 5, 6, 7 and 8 = 840
Hence the number can be written in the form (840k + 3) which is divisible by 9.
If k = 1, number = (840 × 1) + 3 = 843 which is not divisible by 9.
If k = 2, number = (840 × 2) + 3 = 1683 which is divisible by 9.
Hence 1683 is the least number which when divided by 5, 6, 7 and 8 leaves a remainder 3,
but when divided by 9 leaves no remainder.

১,৪০৫.

The next number of the sequence 1, 4, 14, 45, 139, ...... is

ক
ক) 456
খ
খ) 430
গ
গ) 422
ঘ
ঘ) 410

ব্যাখ্যা

প্রশ্ন : The next number of the sequence 1, 4, 14, 45, 139, ...... is

সমাধান :

1, 4, 14, 45, 139, .....

পদগুলোকে নিম্নরূপে লেখা যায় :
4 = (1 × 3) + 1
14 = (4 × 3) + 2
45 = (14 × 3) + 3
139 = (45 × 3) + 4

∴ পরবর্তী পদ হবে
(139 × 3) + 5 = 422

১,৪০৬.

Two number differ by 5. If their product is 336, then the sum of the two numbers is

ক
ক) 21
খ
খ) 37
গ
গ) 28
ঘ
ঘ) 52

ব্যাখ্যা

Question: Two number differ by 5. If their product is 336, then the sum of the two numbers is

Solution:
Let the number be x and y.
Then, x - y = 5 and xy = 336
We know,
(x + y)² = (x - y)² + 4xy
= (5)² + (4 × 336)
= 25 + 1344
= 1369
⇒ x + y = √1369
∴ x + y = 37.

১,৪০৭.

The least number by which 1470 must be divided to get a number which is a perfect square is-

ক
ক) 5
খ
খ) 6
গ
গ) 15
ঘ
ঘ) 30

ব্যাখ্যা

1470 = 2 x 3 x 5 x 7 x 7
So, the least number is = 2 x 3 x 5 = 30

১,৪০৮.

What is the greatest prime factor of 25¹⁵ − 5²⁸?

ক
3
খ
5
গ
7
ঘ
11

ব্যাখ্যা

Question: What is the greatest prime factor of 25¹⁵ − 5²⁸?

Solution:
25¹⁵ − 5²⁸= (5²)¹⁵ - 5²⁸
= 5³⁰ - 5²⁸
= 5²⁸(5² - 1)
= 5²⁸(25 -1)
= 5²⁸ × 24
= 5²⁸ × 8 × 3
= 5²⁸ × 2³ × 3

সুতরাং, সম্পূর্ণ রাশিটির মৌলিক গুণনীয়কগুলো (prime factors) হলো 5, 2, এবং 3.
এর মধ্যে সবচেয়ে বড় হল 5.
সুতরাং, বৃহত্তম মৌলিক গুণনীয়ক (prime factors) হলো 5.

১,৪০৯.

If a and b are even numbers, Which number is odd?

ক
ab
খ
a + b + 2
গ
a + b
ঘ
ab + 1

ব্যাখ্যা

Question: If a and b are even numbers, Which number is odd?

Solution:
We know,
even × even = even
even + even + 2 = even
even + even = even
even × even - 1 = odd

১,৪১০.

If n is a positive integer, which of the following must be even?

ক
n + 1
খ
n + 2
গ
2n
ঘ
2n + 1

ব্যাখ্যা

Question: If n is a positive integer, which of the following must be even?

Solution:
Since n is an integer, it can be EVEN or ODD.
But any integer when multiplied by 2 is even.
For example, 3 × 2 = 6 (even)

The rule is:
EVEN × ODD = EVEN
EVEN × EVEN = EVEN
ODD × ODD = ODD

A) we cant say as it depends if n is odd or even

B) same as (A)

C) Even as n is multiplied by 2

D) It is ODD as EVEN+1 = ODD

E) depends if n is even or odd.

১,৪১১.

The product of two numbers is 300, and the sum of their squares is 625. What is the sum of two numbers?

ক
33
খ
34
গ
35
ঘ
36

ব্যাখ্যা

Question: The product of two numbers is 300, and the sum of their squares is 625. What is the sum of two numbers?

Solution:
Let the numbers be x and y.
As per the question:
xy = 300
x² + y² = 625

So,
(x + y)² = x² + y² + 2xy
= 625 + 2 × 300
= 625 + 600
= 1225

∴ x + y = √1225 = 35

১,৪১২.

Let N be the smallest positive integer that is divisible by both 12 and 15. How many distinct prime factors does N have?

ক
2
খ
3
গ
4
ঘ
5
ঙ
None of these

ব্যাখ্যা

Question: Let N be the smallest positive integer that is divisible by both 12 and 15. How many distinct prime factors does N have?

Solution:
এখানে, N হলো 12 এবং 15 দ্বারা বিভাজ্য ক্ষুদ্রতম সংখ্যা।

সুতরাং, N হবে 12 এবং 15 এর ল.সা.গু।

এখন, 12 = 2 × 2 × 3 = 2² ×3

এবং 15 = 3 × 5

LCM(12,15) = 2² × 3 × 5
= 60

অতএব, N = 60
60 এর মৌলিক উৎপাদক = 2² × 3 × 5

স্বতন্ত্র মৌলিক উৎপাদকগুলি হলো 2, 3 এবং 5।

∴ N এর স্বতন্ত্র মৌলিক উৎপাদকের সংখ্যা হলো 3টি।

১,৪১৩.

The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. Their sum is-

ক
20
খ
38
গ
32
ঘ
26

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. Their sum is-

Solution:
Let the numbers are a, b and c.
Then,
a² + b² + c² = 138
and ab + bc + ca = 131.

Now,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca) = 138 + (2 × 131) = 138 + 262 = 400
⇒ (a + b + c)² = 400
∴ a + b + c = 20.

১,৪১৪.

28 is divided into two parts such that 6 times the first part added to 4 times the second part makes 152. The first part is-

ক
ক) 16
খ
খ) 18
গ
গ) 20
ঘ
ঘ) 22

ব্যাখ্যা

Let, first number be a and second number be b
Here, a+b = 28
ATQ, 6a+4b=152
⇒ 6a+4(28-a) = 152
⇒ 6a + 112 - 4a = 152
⇒ 2a = 152 - 112 = 40
⇒ a = 20

১,৪১৫.

What is the sum of first 17 terms of an AP, if the 1st term is - 20 and last term is 28?

ক
62
খ
64
গ
67
ঘ
68
ঙ
69

ব্যাখ্যা

First term of AP = a = -20 and last term = l = 28
Number of terms = n = 17
Sum of AP = n/2 × (a+l)=(17/2) × (−20+28) = 17×4 = 68

১,৪১৬.

How many integers from 1 to 1000 are divisible by 16 but not by 30?

ক
50
খ
56
গ
58
ঘ
62

ব্যাখ্যা

প্রশ্ন: How many integers from 1 to 1000 are divisible by 16 but not by 30?

সমাধান:
৩০ ও ১৬ এর ল.সা.গু = ২৪০

১০০০ ÷ ২৪০ = ভাগফল ৪, ভাগশেষ ৪০
৩০ ও ১৬ উভয় সংখ্যা দ্বারা ১ থেকে ১০০০ এর মধ্যে বিভাজ্য পূর্ণসংখ্যা ৪টি

১০০০ ÷ ১৬ = ভাগফল ৬২, ভাগশেষ ৮
১৬ দ্বারা ১ থেকে ১০০০ এর মধ্যে বিভাজ্য পূর্ণসংখ্যা ৬২টি

১ থেকে ১০০০ এর মধ্যে ১৬ দ্বারা বিভাজ্য কিন্তু ৩০ দ্বারা বিভাজ্য নয় এমন সংখ্যা ৬২ - ৪ টি
= ৫৮টি

১,৪১৭.

What is the value of a, if 3x² + ax + a + 3 divisible by x + 2?

ক
ক) 15
খ
খ) - 15
গ
গ) - 9
ঘ
ঘ) 9

ব্যাখ্যা

ধরি, f(x) = 3x² + ax + a +3
যেহেতু, x + 2 দ্বারা বিভাজ্য সেহেতু,
f(-2) = 0 হবে
⇒ 3(-2)² + a(-2) + a + 3 = 0
⇒ 12 - 2a + a + 3 = 0
⇒ - a + 15 = 0
⇒ a = 15

১,৪১৮.

Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:

ক
15
খ
29
গ
36
ঘ
12
ঙ
None of the above

ব্যাখ্যা

Question: Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:

Solution:
Let the three odd integers be x, x + 2 and x + 4.

Then,
3x = 2(x + 4) + 3
⇒ 3x = 2x + 8 + 3
∴ x = 11

Third integer = x + 4 = 11 + 4 = 15

১,৪১৯.

ক
0.9
খ
0.3
গ
0.03
ঘ
0.008

ব্যাখ্যা

Question:

Solution:

১,৪২০.

A number is multiplied by 5 and then 7 is added. The result is 52. What is the number?

ক
7
খ
13
গ
6
ঘ
9

ব্যাখ্যা

Question: A number is multiplied by 5 and then 7 is added. The result is 52. What is the number?

Solution:
Let the number be = x

According to the question,
5x + 7 = 52
⇒ 5x = 52 - 7
⇒ 5x = 45
⇒ x = 45/5
⇒ x = 9

১,৪২১.

If a number is in the form of 8⁵ × 9⁷ × 7⁶, find the total number of prime factors of the given number.

ক
ক) 30
খ
খ) 32
গ
গ) 35
ঘ
ঘ) 52

ব্যাখ্যা

The number 8⁵× 9⁷ × 7⁶ can be written as (2³)⁵ × (3²)⁷ × 7⁶The number can be written as 2¹⁵× 3¹⁴× 7⁶
Total number of prime factors = 15 + 14 + 6
∴ The total number of prime factors are 35

১,৪২২.

What will be the least number which when doubled will be exactly divisible by 18, 24, 28, and 36?

ক
1008
খ
504
গ
360
ঘ
252

ব্যাখ্যা

Question: What will be the least number which when doubled will be exactly divisible by 18, 24, 28, and 36?

Solution:
LCM of 18, 24, 28, and 36 is = 504
So, the number will be half of 504 = 504/2 = 252

১,৪২৩.

If the sum of the squares three consecutive natural numbers is 110, then the smallest of these natural numbers is = ?

ক
ক) 5
খ
খ) 6
গ
গ) 7
ঘ
ঘ) 8

ব্যাখ্যা

Let the three consecutive numbers are x, x + 1, x + 2
According to question
x² + (x + 1)² + (x + 2)² = 110
⇒ x² + x² + 1 + 2x + x² + 4 + 4x = 110
⇒ 3x² + 6x + 5 = 110
⇒ 3x² + 6x - 105 = 0
⇒ x² + 2x - 35 = 0
⇒ x^₂ + 7x - 5x - 35 = 0
⇒ x(x + 7) -5(x + 7) = 0
⇒ (x + 7)(x - 5) = 0
⇒ x = 5 & -7
[∴ (-) value can not considered]
∴ Smallest number is = 5

১,৪২৪.

Which of the following is a perfect square?

ক
45
খ
72
গ
49
ঘ
81.5

ব্যাখ্যা

Question: Which of the following is a perfect square?

Solution:
বর্গসংখ্যা (perfect square): সাধারণভাবে একটি স্বাভাবিক সংখ্যা m কে যদি অন্য একটি স্বাভাবিক সংখ্যা n এর বর্গ (n²) আকারে প্রকাশ করা যায় তবে m বর্গসংখ্যা।
m সংখ্যাগুলোকে পূর্ণবর্গসংখ্যা বলা হয়।

পূর্ণবর্গ সংখ্যার ধর্ম:
• যে সংখ্যার সর্ব ডানদিকের অঙ্ক অর্থাৎ একক স্থানীয় অঙ্ক ২ বা ৩ বা ৭ বা ৮ তা পূর্ণবর্গ নয় ।
• যে সংখ্যার শেষে বিজোড় সংখ্যক শূন্য থাকে, ঐ সংখ্যা পূর্ণবর্গ নয়।
• একক স্থানীয় অঙ্ক ১ বা ৪ বা ৫ বা ৬ বা ৯ হলে, ঐ সংখ্যা পূর্ণবর্গ হতে পারে। যেমন : ৮১, ৬৪, ২৫, ৩৬, ৪৯ ইত্যাদি বর্গসংখ্যা ।
• আবার সংখ্যার ডানদিকে জোড়সংখ্যক শূন্য থাকলে ঐ সংখ্যা পূর্ণবর্গ হতে পারে। যেমন : ১০০, ৪৯০০ ইত্যাদি বর্গসংখ্যা ।

প্রদত্ত অপশগুলোর মধ্যে
গ) 49 = 7² [যা পূর্ণবর্গসংখ্যা]

১,৪২৫.

Two baskets together have 640 oranges. If one-fifth of the oranges in the first basket be taken to the second basket then, numbers of oranges in both baskets become equal. The number of oranges in the first basket is-

ক
ক) 800
খ
খ) 600
গ
গ) 400
ঘ
ঘ) 300

ব্যাখ্যা

Let the number of oranges in first basket be x,
Number of oranges in second basket = 640 - x
ATQ, x - x/5 = 640 - x + x/5
⇒ 4x/5 = 640 - 4x/5
⇒ 4x/5 + 4x/5 = 640
⇒ 8x/5 = 640
⇒ x = 640 × (5/8)
⇒ x = 400
∴ Number of oranges in first basket = 400.

১,৪২৬.

The difference between a number and its three-fifths is 240, What is the number?

ক
ক) 125
খ
খ) 300
গ
গ) 420
ঘ
ঘ) 600

ব্যাখ্যা

Question: The difference between a number and its three-fifths is 240, What is the number?

Solutionh:
let the number x

x - 3x/5 = 240
⇒ (5x - 3x)/5 = 240
⇒ 2x = 1200
∴ x = 600

১,৪২৭.

If n is an integer between 20 and 80, then any of the following could be n + 7 except-

ক
88
খ
80
গ
47
ঘ
56

ব্যাখ্যা

Question: If n is an integer between 20 and 80, then any of the following could be n + 7 except-

Solution:
maximum value of n + 7 = 80 + 7 = 87
88 > 87
So, the correct answer is A

১,৪২৮.

The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is:

ক
ক) 276
খ
খ) 299
গ
গ) 322
ঘ
ঘ) 345

ব্যাখ্যা

Clearly, the numbers are (23 x 13) and (23 x 14).
Larger number = (23 x 14) = 322.

১,৪২৯.

1 বিলিয়ন = কত কোটি?

ক
10
খ
100
গ
1000
ঘ
10000

ব্যাখ্যা

প্রশ্ন: 1 বিলিয়ন = কত কোটি?

সমাধান:
• 1 বিলিয়ন = 100 কোটি

• 1 মিলিয়ন = 10 লক্ষ
• 10 মিলিয়ন = 1 কোটি
• 1000 বিলিয়ন = 1 ট্রিলিয়ন
• 100000 কোটি বা, 1 লক্ষ কোটি = 1 ট্রিলিয়ন

১,৪৩০.

When 4 is added to 1/2 of a number, the result is 14. What is the number?

ক
10
খ
16
গ
20
ঘ
24

ব্যাখ্যা

Question: When 4 is added to 1/2 of a number, the result is 14. What is the number?

Solution:
let the number be = x

ATQ,
(x/2) + 4 = 14
⇒ x/2 = 10
∴ x = 20

১,৪৩১.

3.003/2.002 = ?

ক
ক) 1.05
খ
খ) 1.50015
গ
গ) 1, 501
ঘ
ঘ) 1.5015
ঙ
ঙ) 1.5

ব্যাখ্যা

Question: 3.003 / 2.002 =?

Solution:
3.003 / 2.002
= 3003/2002
= (2002 + 1001)/2002
= (2002/2002) + (1001/2002)
= 1 + (1/2)
= 1 + 0.5
= 1.5

১,৪৩২.

Which of the following is prime number?

ক
ক) 143
খ
খ) 289
গ
গ) 359
ঘ
ঘ) 117

ব্যাখ্যা

Question: Which of the following is prime number?

Solution:
১৪৩ সংখ্যাটি ১১ দ্বারা বিভাজ্য।
২৮৯ সংখ্যাটি ১৭ দ্বারা বিভাজ্য।
১১৭ সংখ্যাটি ৩ দ্বারা বিভাজ্য।
৩৫৯ সংখ্যাটি ১ ও ৩৫৯ ছাড়া অন্য কোনো সংখ্যা দ্বারা বিভাজ্য নয়। তাই ৩৫৯ একটি মৌলিক সংখ্যা।

১,৪৩৩.

একটি সংখ্যাকে ৮৪০ এবং ৭৬০ এর যোগফল দ্বারা ভাগ করলে ভাগফল, দুটি সংখ্যার পার্থক্যের তিনগুণ হয় এবং ভাগশেষ ৪৫ পাওয়া যায়। সংখ্যাটি কত?

ক
৩৮৪,০৪৫
খ
৫৬০০১
গ
৬৫৭১৯০
ঘ
২৬৮২২
ঙ
কোনোটিই নয়

ব্যাখ্যা

প্রশ্ন: একটি সংখ্যাকে ৮৪০ এবং ৭৬০ এর যোগফল দ্বারা ভাগ করলে ভাগফল, দুটি সংখ্যার পার্থক্যের তিনগুণ হয় এবং ভাগশেষ ৪৫ পাওয়া যায়। সংখ্যাটি কত?

সমাধান:
যোগফল = (৮৪০ + ৭৬০) = ১৬০০
পার্থক্য = (৮৪০ - ৭৬০) = ৮০

∴ ভাগফল = (৩ × ৮০) = ২৪০

এখন,
সংখ্যাটি = (২৪০ × ১৬০০) + ৪৫
= ৩৮৪,০০০ + ৪৫
= ৩৮৪,০৪৫

১,৪৩৪.

If x and y are consecutive positive integers, which of the following must be an even integer?

ক
ক) x
খ
খ) y
গ
গ) xy/2
ঘ
ঘ) xy

ব্যাখ্যা

Question: If x and y are consecutive positive integers, which of the following must be an even integer?

Solution:
দুইটি ক্রমিক স্বাভাবিক সংখ্যার একটি জোড় হলে অপরটি অবশ্যই বিজোড় হবে।

একটি জোড় এবং একটি বিজোড় পূর্ণ সংখ্যার গুণফল অবশ্যই একটি জোড় পূর্ণসংখ্যা হবে।

তাই, x এবং y দুইটি ক্রমিক সংখ্যা হলে xy জোড় হবে।

১,৪৩৫.

Rohit multiplies a number by 2 instead of dividing the number by 2. Resultant number is what percentage of the correct value?

ক
200%
খ
300%
গ
50%
ঘ
400%

ব্যাখ্যা

Question: Rohit multiplies a number by 2 instead of dividing the number by 2. Resultant number is what percentage of the correct value?

Solution:
Let the number be x.
Correct value = x/2
Resultant number = 2 × x

According to Question,
Required percentage = Resultant number/Correct value × 100%
⇒ Required percentage = [(2x)/(x/2)] × 100%
⇒ Required percentage = 400%

১,৪৩৬.

The next number in the sequence 4, 5, 8, 17, 33, … … is

ক
57
খ
59
গ
58
ঘ
63

ব্যাখ্যা

The series is:
4 + 0² = 4,
4 + 1² = 5,
4 + 2² = 8,
8 + 3² = 17,
17 + 4² = 33,
33 + 5² = 58

১,৪৩৭.

Mr. A has won an election by a vote of 250 to 150. What part of the total vote was against him?

ক
ক) 2/5
খ
খ) 3/5
গ
গ) 4/7
ঘ
ঘ) 7/15
ঙ
ঙ) 3/8

ব্যাখ্যা

Question: Mr. A has won an election by a vote of 250 to 150. What part of the total vote was against him?

Solution:
Total votes 250 + 150 = 400

total vote was against him 150/400 = 3/8

১,৪৩৮.

Which of the fractions given below is the largest?

ক
5/4
খ
6/5
গ
7/6
ঘ
4/3

ব্যাখ্যা

Question: Which of the fractions given below is the largest?

Solution:
Each fraction is equivalent to a decimal
4/3 = 1.33
5/4 = 1.25
6/5 = 1.2
7/6 = 1.167

Hence, 4/3 is the largest fraction.

১,৪৩৯.

The average of two numbers is 62. If 2 is added to the smaller number, the ratio between the numbers becomes 1 : 2. The difference between two numbers are -

ক
84
খ
40
গ
34
ঘ
44

ব্যাখ্যা

Question: The average of two numbers is 62. If 2 is added to the smaller number, the ratio between the numbers becomes 1 : 2. The difference between two numbers are -

Solution:
Let,
the small number = x
Then, the large number = 124 - x

According to the question,
(x + 2)/(124 - x) = 1/2
⇒ 2x + 4 = 124 - x
⇒ 2x + x = 124 - 4
⇒ 3x = 120
∴ x = 40

So, the Small number = 40
The large number = 124 - 40
= 84

∴ Difference between two numbers = 84 - 40
= 44

১,৪৪০.

Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:

ক
ক) 9
খ
খ) 11
গ
গ) 13
ঘ
ঘ) 15

ব্যাখ্যা

প্রশ্ন: Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:

সমাধান:
Let the three odd integers be x, x + 2 and x + 4.

Then,
3x = 2(x + 4) + 3
⇒ 3x = 2x + 8 + 3
∴ x = 11

Third integer = x + 4 = 11 + 4 = 15

১,৪৪১.

How many number between 1000 and 5000 are exactly divisible by 225 ?

ক
17
খ
18
গ
19
ঘ
20

ব্যাখ্যা

Question: How many numbers between 1000 and 5000 are exactly divisible by 225?

Solution:
225 × 5 = 1125
.
.
.
225 × 22 = 4950

numbers between 1000 and 5000 are exactly divisible by 225 = 22 - 5 + 1
= 18

১,৪৪২.

On dividing a number by 5, we get 3 as remainder. What will be the remainder when the square of this number is divided by 5?

ক
0
খ
1
গ
2
ঘ
4
ঙ
None of these

ব্যাখ্যা

প্রশ্ন: On dividing a number by 5, we get 3 as remainder. What will be the remainder when the square of this number is divided by 5?

সমাধান:
Let the number be x and on dividing x by 5, we get m as quotient and 3 as remainder.
∴ x = 5m + 3
⇒ x²= (5m + 3)²
= (25m² + 30m + 9)
= 5(5m² + 6m + 1) + 4

∴ On dividing x² by 5, we get 4 as remainder.

১,৪৪৩.

Find the greatest number that will divide 45, 93 and 185 so as to leave the same remainder in each case.

ক
ক) 5
খ
খ) 4
গ
গ) 3
ঘ
ঘ) 2

ব্যাখ্যা

Question: Find the greatest number that will divide 45, 93 and 185 so as to leave the same remainder in each case.

Solution:
Here,
93 - 45 = 48
185 - 93 = 92
185 - 45 = 140

So, the required number = H.C.F of 48, 92 and 140
= 4

১,৪৪৪.

In a seminar, the number of participants in Bangla, English and Mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required, where in each room the same number of participants are to be seated; and all of them being in the same subject.

ক
20
খ
21
গ
22
ঘ
23

ব্যাখ্যা

Question: In a seminar, the number of participants in Bangla, English and Mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required, where in each room the same number of participants are to be seated; and all of them being in the same subject.

Solution:
HCM of 60, 84 and 108 = 12

Minimum number of rooms required = (60/12) + (84/12) + (108/12)
= 5 + 7 + 9
= 21

১,৪৪৫.

What will be the least number which when halved will be exactly divisible by 12, 18, 21 and 30?

ক
ক) 930
খ
খ) 1800
গ
গ) 2520
ঘ
ঘ) 3010

ব্যাখ্যা

Question: What will be the least number which when halved will be exactly divisible by 12, 18, 21 and 30?

Solution:
12 = 2 × 2 × 3
18 = 2 × 3 × 3
21 = 3 × 7
30 = 2 × 3 × 5

L.C.M. of 12, 18, 21 30 = 2 × 3 × 2 × 3 × 7 × 5 = 1260
Required number = 1260 × 2 = 2520

১,৪৪৬.

What is the sum of two consecutive even numbers, the difference of whose squares is 84?

ক
ক) 34
খ
খ) 38
গ
গ) 42
ঘ
ঘ) 46

ব্যাখ্যা

Let the numbers be x and x + 2.
Then, (x + 2)² - x² = 84
⇒ 4x + 4 = 84
⇒ 4x = 80
⇒ x = 20.
∴ The required sum
= x + (x + 2)
= 2x + 2
= 42

১,৪৪৭.

The ratio between a two-digit number and the sum of the digit of that number is 4 : 1. If the digit in the unit’s place is 3 more than the digit in the ten’s place, what is the number.

ক
ক) 27
খ
খ) 36
গ
গ) 42
ঘ
ঘ) 48

ব্যাখ্যা

Let the ten’s digit be x.
Then, unit’s digit = (x+3).
Sum of the digits = x + (x +3) = 2x + 3.
Number = 10x + (x+3) = 11x + 3.
(11x + 3)/(2x + 3) = 4/1
11x + 3 = 8x + 12
11x - 8x = 12 - 3
3x = 9
x = 3
Hence, Required number = 11x + 3 = 36.

১,৪৪৮.

ক
9
খ
5
গ
13
ঘ
11

ব্যাখ্যা

Problem:

Solution :

১,৪৪৯.

48.95 - 32.006 =?

ক
ক) 16.089
খ
খ) 16.35
গ
গ) 16.944
ঘ
ঘ) 16.89

ব্যাখ্যা

Question: 48.95 - 32.006 =?

Solution:
48.95 - 32.006
= 16.944

১,৪৫০.

The sum of the numbers is 184. If one-third of the one exceeds one-seventh of the other by 8, find the smaller number.

ক
ক) 72
খ
খ) 44
গ
গ) 64
ঘ
ঘ) 40

ব্যাখ্যা

Let the numbers be x and (184-x).
Then,
x/3 - (184 -x)/7 = 8
⇒ 7x – 3(184-x) = 168
⇒ 10x = 720
⇒ x = 72.
So, the numbers are 72 and 112.
Hence, a smaller number = 72.

১,৪৫১.

In which one of the following choices, p must be greater than q?

ক
ক) 0.9^p = 0.9^q
খ
খ) 0.9^p = 0.9^2q
গ
গ) 9^p < 9^q
ঘ
ঘ) 9^p > 9^q

ব্যাখ্যা

অপশন (ক) হতে আমরা পাই,
0.9^p = 0.9^q
p = q
অপশন (খ) হতে আমরা পাই,
0.9^p = 0.9^2q
p = 2q
এখানে, p ও q ধনাত্মক হলে p > q অর্থাৎ p = 2 ও q
= 1 হলে p = 2q এবং p > q সত্য।
কিন্তু p ও q ঋণাত্মক হলে
অর্থাৎ p = - 2 ও q = - 1 হলে p = 2q কিন্তু p > q এ জন্য অপশন খ অনুযায়ী p > q সর্বদা সত্য হয় না।
অপশন (গ) হতে আমরা পাই. p <q
অপশন (ঘ) হতে আমরা পাই, p > q

অপশন (ঘ) সঠিক

১,৪৫২.

If the numerator of a fraction is increased by 2 and the denominator by 3 it becomes 1. Again, if the numerator decreased by 4 and the denominator by 2 it becomes 1/2. Find the fraction.

ক
ক) 7/8
খ
খ) 5/4
গ
গ) 6/7
ঘ
ঘ) 4/5

ব্যাখ্যা

প্রশ্ন : If the numerator of a fraction is increased by 2 and the denominator by 3 it becomes 1. Again, if the numerator decreased by 4 and the denominator by 2 it becomes 1/2. Find the fraction.

সমাধান :
মনে করি, লব = x
হর = y
ভগ্নাংশ = x/y
(x +2)/(y + 3) = 1
⇒ x +2 = y + 3
⇒ x - y = 1 ...................(1)

(x - 4)/(y - 2) = 1/2
⇒ 2x - 8 = y - 2
⇒ 2x - y = 6 .................(2)
(2) থেকে (1) বিয়োগ করে পাই,
x = 5

x এর মান (1) নং এ বসিয়ে পাই,
y = 5 - 1 = 4

∴ ভগ্নাংশ = 5/4

১,৪৫৩.

The least perfect square, which is divisible by each of 21, 36 and 66 is:

ক
ক) 213444
খ
খ) 214344
গ
গ) 214434
ঘ
ঘ) 231444

ব্যাখ্যা

L.C.M. of 21, 36, 66 = 2772.
Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11
To make it a perfect square, it must be multiplied by 7 x 11.
So, required number = 2² x 3² x 7² x 11² = 213444

১,৪৫৪.

A water filter can be filled with 8 jugs of capacity 1.3 liters each. How many jugs are required to fill the same filter, if the capacity of the jug is 0.8 liters?

ক
ক) 15
খ
খ) 12
গ
গ) 8
ঘ
ঘ) 13
ঙ
ঙ) None of these

ব্যাখ্যা

প্রশ্নমতে ফিল্টারেরে ধারণক্ষমতা 8×1.3 = 10.4
তাহলে এই ফিল্টারের জন্য 0.8 লিটারের ধারণক্ষমতার জগ প্রয়োজন = 10.4/0.8 = 13 টি।

১,৪৫৫.

Two numbers when divided by a certain divisor give remainder 35 and 30 respectively and when their sum is divided by the same divisor, the remainder is 20, then the divisor is :

ক
ক) 40
খ
খ) 45
গ
গ) 50
ঘ
ঘ) 55

ব্যাখ্যা

The divisor should be (35 + 30) - 20 = 45
Reason:
Let us say the dividends are A and B and the common divisor be d
A = d*p + 35 … (1)
B = d*q + 30 … (2) [ p & q are respective quotients]
A + B=d(p + q) + 35 + 30
A + B = d(p + q) + 65
now since remainder is 20
A + B = d(p + q) + 45 + 20
So, the divisor must be 45

১,৪৫৬.

Given that √24 is approximately equal to 4.898, √(8/3) is nearly equal to-

ক
0.644
খ
1.633
গ
2.5
ঘ
None of these

ব্যাখ্যা

Question: Given that √24 is approximately equal to 4.898, √(8/3) is nearly equal to-

Solution:
√(8/3)
=√{(8 × 3)/(3 × 3)}
= √24/3
= 4.898/3
= 1.633

১,৪৫৭.

The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is-

ক
380
খ
395
গ
400
ঘ
425

ব্যাখ্যা

Question: The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is-

Solution:
Let the numbers be x and y. [x > y]
Then, xy = 9375 and
x/y = 15

(xy)/(x/y) = 9375/15
⇒ y² = 625
∴ y = 25

Now,
x = 15y = (15 × 25) = 375.
Sum of the numbers = x + y = 375 + 25 = 400

১,৪৫৮.

If both x and y are prime numbers, which of the following CANNOT be the product of x and y?

ক
ক) 6
খ
খ) 10
গ
গ) 35
ঘ
ঘ) 27

ব্যাখ্যা

6 = 2 × 3
10 = 3 × 5
35 = 5 × 7
27 = 3 × 9; Here 9 is not a prime number

১,৪৫৯.

If the average of 5 consecutive integers is 15 then what is the difference between least and the greatest of the 5 integers?

ক
4
খ
5
গ
8
ঘ
10
ঙ
None of these

ব্যাখ্যা

প্রশ্ন: If the average of 5 consecutive integers is 15 then what is the difference between least and the greatest of the 5 integers?

সমাধান:
ধরি, পাঁচটি ক্রমিক সংখ্যা x, x + 1, x + 2, x + 3, x + 4

পাঁচটি ক্রমিক সংখ্যার গড় = 15
পাঁচটি সংখ্যার সমষ্টি = 15 × 5 = 75

প্রশ্নমতে,
x + x + 1 + x + 2 + x + 3 + x + 4 = 75
⇒ 5x + 10 = 75
⇒ 5x = 65
∴ x = 13

সবচেয়ে ছোট সংখ্যাটি = 13
সবচেয়ে বড় সংখ্যাটি = 13 + 4 = 17

∴ সবচেয়ে বড় সংখ্যা ও সবচেয়ে ছোট সংখ্যার পার্থক্য = 17 - 13 = 4

১,৪৬০.

There are 48 books on 3 shelves. If 3 books from third shelf are shifted to the second there will be same number of books on first and third shelves and double the number of books on second shelf. How many books were there on three shelves originally?

ক
ক) 10, 25,13
খ
খ) 12, 21,15
গ
গ) 11, 23, 14
ঘ
ঘ) 13, 19, 16
ঙ
ঙ) None

ব্যাখ্যা

Suppose, the first shelf had x books.
So, the third shelf had x+3 books.
And the second shelf had 2x-3 books.
According to the question,
x + x+3 + 2x-3 = 48
4x = 48
x = 12
1st shelf = 12, 2nd shelf = 2×12-3 = 21, 3rd shelf = 12+3 = 15.

১,৪৬১.

A number is doubled and 9 is added. If the resultant is tripled, it becomes 81. What is that number?

ক
ক) 8
খ
খ) 9
গ
গ) 10
ঘ
ঘ) 11

ব্যাখ্যা

Question: A number is doubled and 9 is added. If the resultant is tripled, it becomes 81. What is that number?

Solution:
Let
The number be x

Now
3(2x + 9) = 81
⇒ 6x + 27 = 81
⇒ 6x = 81 - 27
⇒ 6x = 54
⇒ x = 54/6
⇒ x = 9

∴ The number is 9

১,৪৬২.

The difference between two numbers is 5 and the difference between their squares is 65. What is the larger number?

ক
ক) 13
খ
খ) 11
গ
গ) 8
ঘ
ঘ) 9

ব্যাখ্যা

প্রশ্ন: The difference between two numbers is 5 and the difference between their squares is 65. What is the larger number?

সমাধান:
ধরি,
বৃহত্তম সংখ্যাটি = x
ক্ষুদ্রতম সংখ্যাটি = y

প্রশ্নমতে,
x - y = 5 ..............(1)
x² - y² = 65 ..............(2)

(2)নং সমীকরণ হতে পাই,
x² - y² = 65
(x + y)(x - y) = 65
(x + y) × 5 = 65
x + y = 13.................(3)

(1) + (3)⇒
x - y + x + y = 5 + 13
2x = 18
x = 9

১,৪৬৩.

The product of three consecutive natural numbers is always divisible by

ক
ক) 6
খ
খ) 7
গ
গ) 13
ঘ
ঘ) 11

ব্যাখ্যা

Let n, n + 1, n + 2 be three consecutive natural numbers and P(n) be their product.

We have,
n= 1 ⇒ n(n + 1)(n + 2) =1 × 2 × 3 = 6, which is divisible by 3 and 6.
n= 2 ⇒ n(n + 1)(n + 2) =2 × 3 × 4 = 24, which is divisible by 3, 8 and 6.
n= 3 ⇒ n(n + 1)(n + 2) =3 × 4 × 5 = 60, which is divisible by 3 and 6.
n= 4 ⇒n(n + 1)(n + 2) =4 × 5 × 6 = 120, which is divisible by 3, 8 and 6.

Hence, n(n + 1)(n + 2) is divisible by 3 and 6 for all n∈ N.

১,৪৬৪.

What is the next number in the series 1, 4, 9, 16.......?

ক
ক) 25
খ
খ) 23
গ
গ) 21
ঘ
ঘ) 31

ব্যাখ্যা

এখানে,
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25

১,৪৬৫.

HCF of two number is 4 and the sum of those two numbers is 36. Find how many such pair of number is possible.

ক
ক) 1
খ
খ) 2
গ
গ) 3
ঘ
ঘ) 4

ব্যাখ্যা

HCF of two number is 4
Let, those numbers be 4x and 4y where x and y are prime to each other Accordingly,
4x + 4y = 36
⇒ 4(x + y) = 36
⇒ (x + y) = 9
Now,
9 = 8 + 1
9 = 7 + 2
9 = 6 + 3
9 = 5 + 4
In all these cases only (8, 1); (7, 2); and (5, 4) are prime to each other.
So, such three pair is possible
∴ Such three pair of number is possible.

To find the pairs we only can take those pair, which are prime to each other.

We can't take those pair, which are not prime to each other.

For example: In this problem, 9 can be represent as 6 + 3 = 9

But, 6 & 3 are not prime to each other.

So, it can't be a possible pair.

১,৪৬৬.

The numbers 2, 3, 5 and x have an average equal to 4. What is the value of x?

ক
ক) 4
খ
খ) 6
গ
গ) 8
ঘ
ঘ) 10

ব্যাখ্যা

(2 + 3 + 5 + x)/4 = 4
Or, 2 + 3 + 5 + x = 4×4
Or, x = 16 - 10 = 6

১,৪৬৭.

The ratio of two numbers is 1 : 3 and their L.C.M is 75. What is the sum of the numbers?

ক
ক) 90
খ
খ) 100
গ
গ) 115
ঘ
ঘ) 120

ব্যাখ্যা

Question: The ratio of two numbers is 1 : 3 and their L.C.M is 75. What is the sum of the numbers?

Solution:
Let the numbers are x and 3x
So, L.C.M of x and 3x is 3x
ATQ,
3x = 75
∴ x = 25
So, the sum of the numbers
= (x + 3x)
= 4x
= 4 × 25
= 100

১,৪৬৮.

The largest prime factor of {³√(2⁴)⁶} - 1 is-

ক
ক) 23
খ
খ) 17
গ
গ) 5
ঘ
ঘ) 3

ব্যাখ্যা

Question: The largest prime factor of {³√(2⁴)⁶} - 1 is-

Solution:
{³√(2⁴)⁶} - 1
= (2⁴)^6/3 - 1
= (2⁴)² - 1
= (2⁴ + 1) (2⁴ - 1)
= 17 × 15
= 17 × 3 × 5

So, the largest prime factor is 17

১,৪৬৯.

P is a prime number and Q is not divisible by P. What will be the L.C.M of P and Q?

ক
PQ/(P-Q)
খ
PQ
গ
P/Q
ঘ
Q/P
ঙ
(P+Q)/(P-Q)

ব্যাখ্যা

অবিভাজ্য কোন সংখ্যাদ্বয়ের ল.সা.গু ঐ সংখ্যাদ্বয়ের গুনফলের সমান।

১,৪৭০.

If n(U) = 50, n(A) = 28, n(B) = 26 and n(A ∩ B) = 12 then what is n(A ∪ B)′ ?

ক
9
খ
12
গ
8
ঘ
10
ঙ
None of these

ব্যাখ্যা

Question: If n(U) = 50, n(A) = 28, n(B) = 26 and n(A ∩ B) = 12 then what is n(A ∪ B)′ ?

Solution:
We know that,
n(A ∪ B)= n(A) + n(B) - (A ∩ B)
= 28 + 26 - 12
= 42

Now,
n(A ∪ B)′= n(U) - n(A ∪ B)
= 50 - 42
= 8

So, n(A ∪ B)′ = 8

১,৪৭১.

If both 5² and 3² are factors of m where m = z × 2⁵ × 6² × 7³ , what is the smallest positive value of z?

ক
9
খ
16
গ
25
ঘ
32
ঙ
36

ব্যাখ্যা

Question: If both 5² and 3² are factors of m where m = z × 2⁵ × 6² × 7³ , what is the smallest positive value of z?

Solution:
Given, m = z × 2⁵ × 6² × 7³

We simplify this by expressing all terms in their prime factorizations except z,
∴ 6² = (2 × 3)² = 2² × 3²
∴ m = z × 2⁵ × 2² × 3² × 7³
= z × 2⁷ × 3² × 7³

If both 5² and 3² are factors, then they must be present in the number.
From the simplified expression of m, we can see that the factor 3² is already present but the required factor 5² is missing.
To make 5² a factor of m, it must come from z.

∴ The smallest positive value of z = 5² = 25.

১,৪৭২.

When 52416 is divided by 312, the quotient is 168. What will be the quotient when 52.416 is divided by 0.0168?

ক
ক) 3.120
খ
খ) 31.20
গ
গ) 3120
ঘ
ঘ) None of these

ব্যাখ্যা

Given,
52416/312=168
⇔ 52416/ 168 =312
Now, 52.416/ 0.0168
= 524160/ 168
= (52416/ 168 )×10
=312×10
=3120

১,৪৭৩.

The smallest prime factor of (2⁴)² − 1 is -

ক
ক) 3
খ
খ) 5
গ
গ) 7
ঘ
ঘ) 9

ব্যাখ্যা

(2⁴)² − 1
= 2⁸ - 1
= 256 - 1
= 255
= 3 × 5 × 17

So, the smallest prime factor is 3

১,৪৭৪.

How many prime numbers are there between 56 and 100?

ক
ক) 8
খ
খ) 9
গ
গ) 10
ঘ
ঘ) 11
ঙ
ঙ) 12

ব্যাখ্যা

Prime numbers between 56 and 100
59, 61, 67, 71, 73, 79, 83, 89, 97 are the prime numbers between 56 and 100.

১,৪৭৫.

The four digit smallest positive number which when divided by 4, 5, 6 or 7, it leaves always the remainder as 3:

ক
1000
খ
1257
গ
1263
ঘ
1683

ব্যাখ্যা

Question: The four digit smallest positive number which when divided by 4, 5, 6 or 7, it leaves always the remainder as 3:

Solution:
The least possible number = (LCM of 4, 5, 6 and 7) + 3 = 420 + 3 = 423
The next higher number is, (420m + 3) now we put a least value of m such that (420m + 3) ≥ 1000
value = 420 × 3 + 3 = 1263
∴ m = 3

১,৪৭৬.

The least number by which 320 must be multiplied to make it perfect square, is-

ক
5
খ
25
গ
10
ঘ
15

ব্যাখ্যা

Question: The least number by which 320 must be multiplied to make it perfect square, is-

Solution:
Here,
320 = 2 × 2 × 2 × 2 × 2 × 2 × 5 = 2⁶ × 5
To make it perfect squre, it must be multiplied by 5

Therefore, the least number by which 320 must be multiplied to make it a perfect square is 5.

১,৪৭৭.

Solve for x if 16x-4 = 68+7x.

ক
ক) x = 16
খ
খ) x = -16
গ
গ) x = 8
ঘ
ঘ) x = 5

ব্যাখ্যা

16x - 4 = 68 + 7x
বা, 9x = 72
বা, x = 8.

১,৪৭৮.

If a, b, and c are three consecutive odd integers such that 10 < a < b < c < 20 and if b and c are prime numbers, what is the value of a + b?

ক
32
খ
30
গ
28
ঘ
36

ব্যাখ্যা

Question: If a, b, and c are three consecutive odd integers such that 10 < a < b < c < 20 and if b and c are prime numbers, what is the value of a + b?

Solution:
Consecutive odd integers between 10 and 20 are 11,13,15,17,19
b and c are prime numbers.
Since a is at least 11, then b and c must be 17 and 19.

Hence a = 15, b = 17, c = 19.
∴ a + b = 32

১,৪৭৯.

An apple costs 7 taka each. An orange costs 5 taka each. Rasel spends 38 taka on these fruits. The number apple purchased is-

ক
2
খ
3
গ
4
ঘ
Can't be determined

ব্যাখ্যা

Question: An apple costs 7 taka each. An orange costs 5 taka each. Rasel spends 38 taka on these fruits. The number apple purchased is-

Solution:
ধরি, x টি আপেল ও y টি কমলা কিনেছে।

প্রশ্নমতে,
7x + 5y = 38
⇒ 5y = (38 - 7x)
∴ y = (38 - 7x)/5

x = 2, y= (38 - 14)/5 = 24/5; যা পূর্ণসংখ্যা নয়।
x = 3, y= (38 - 21)/5 = 17/5; যা পূর্ণসংখ্যা নয়।
x = 4, y= (38 - 28)/5 = 10/5 = 2; যা পূর্ণসংখ্যা।

১,৪৮০.

The sum of three consecutive even integers is 54. Find the least integer among them.

ক
16
খ
14
গ
12
ঘ
18

ব্যাখ্যা

Question: The sum of three consecutive even integers is 54. Find the least integer among them.

Solution:
Let us consider three consecutive even integers be x, x + 2 and x + 4

According to the question,
x + (x + 2) + (x + 4) = 54
⇒ 3x + 6 = 54
⇒ 3x = 48
⇒ x = 16

Hence, the least integer = 16

১,৪৮১.

Find a number such that when 15 is subtracted from 7 times the number, the result is 10 more than twice the number.

ক
ক) 5
খ
খ) 10
গ
গ) 15
ঘ
ঘ) 20

ব্যাখ্যা

Let, the number be z,
Then, 7z – 15 = 2z + 10
⇒ 5z = 25
⇒ z = 5.
Hence, the required number is 5.

১,৪৮২.

A student loses 1 mark for every wrong answer and scores 2 marks for every correct answer. If he answers all the 60 questions in an exam and scores 39 marks, how many of them were correct?

ক
ক) 33
খ
খ) 31
গ
গ) 27
ঘ
ঘ) 37

ব্যাখ্যা

Question: A student loses 1 mark for every wrong answer and scores 2 marks for every correct answer. If he answers all the 60 questions in an exam and scores 39 marks, how many of them were correct?

Solution:
ধরি,
ক সংখ্যক উত্তর ভুল দিয়েছে।

যদি ৬০ টি উত্তর সঠিক হতো তাহলে পেত = ৬০ × ২ = ১২০ নম্বর

প্রতিটি ভুল উত্তরের জন্য প্রকৃতপক্ষে কাটা যায় = ২ + ১ = ৩ নম্বর

প্রশ্নমতে,
১২০ - ৩ক = ৩৯
৩ক = ১২০ - ৩৯
ক = ৮১/৩
ক = ২৭

সুতরাং, ২৭ টি উত্তর ভুল ছিল এবং (৬০ - ২৭) বা, ৩৩ টি উত্তর সঠিক ছিল।

১,৪৮৩.

In a farm there are cows and hens. If heads are counted there are 180, if legs are counted there are 420. The number of cows in the farm is -

ক
ক) 150
খ
খ) 30
গ
গ) 60
ঘ
ঘ) 130

ব্যাখ্যা

Question: In a farm there are cows and hens. If heads are counted there are 180, if legs are counted there are 420. The number of cows in the farm is -

Solution:
Let, the number of cows be x.
Then, number of hens would be (180 - x)

ATQ,
(180 - x) × 2 + 4x = 420
⇒ 360 - 2x + 4x = 420
⇒ 2x = 420 - 360
⇒ 2x = 60
⇒ x = 60/2
∴ x = 30

∴ The number of cows 30.

১,৪৮৪.

The sum of four consecutive even integers is 68. What is the value of the largest number?

ক
ক) 16
খ
খ) 18
গ
গ) 20
ঘ
ঘ) 22

ব্যাখ্যা

Let, the numbers be = x, x+2, x+4, x+6
ATQ, x + x + 2 + x + 4 + x + 6 = 68
Or, 4x = 68 - 12 = 56
Or, x = 14

So, the largest number is = 14 + 6 = 20

১,৪৮৫.

If the product of two positive number is 1575 and the ratio is 7:9 then the greatest number is:

ক
45
খ
135
গ
35
ঘ
63
ঙ
None of the above

ব্যাখ্যা

Let the numbers are 7x and 9x

According to the question,
⇒ 7x × 9x = 1575
⇒ 63x² = 1575
⇒ x² = 25
⇒ x = 5

Then greater number
= 9x
= 9 × 5
= 45

১,৪৮৬.

What is the square root of 0.16?

ক
0.2
খ
0.4
গ
1.4
ঘ
2.4

ব্যাখ্যা

Question: What is the square root of 0.16?

Solution:
√0.16
= √(16/100)
= 4/10
= 0.4

১,৪৮৭.

A number is 5 more than its one-sixth part. The number is-

ক
30
খ
6
গ
36
ঘ
5

ব্যাখ্যা

Question: A number is 5 more than its one-sixth part. The number is-

Solution:
Given that,
A number is 5 more than its one-sixth part.
Let the number be x.

According to the question,
x = (1/6)x + 5
⇒ x - (1/6)x = 5
⇒ (6x - x)/6 = 5
⇒ 5x/6 = 5
⇒ x = 5 × (6/5)
∴ x = 6

So the number is 6.

১,৪৮৮.

The next term of the series 9 + 36 + 81 + 144 + ____ is

ক
ক) 169
খ
খ) 256
গ
গ) 272
ঘ
ঘ) 225

ব্যাখ্যা

Given, 9 + 36 + 81 + 144
The series is = 3² + 6² + 9² + 12² + 15²
So, next term is 15² = 225

১,৪৮৯.

The total of three consecutive multiples of 3 is 396. Determine the largest number.

ক
140
খ
145
গ
135
ঘ
162

ব্যাখ্যা

Question: The total of three consecutive multiples of 3 is 396. Determine the largest number.

Solution:
Let,
First multiple: 3x
Second multiple: 3(x + 1) = 3x + 3
Third multiple: 3(x + 2) = 3x + 6

ATQ,
3x + (3x + 3) + (3x + 6) = 396
⇒ 9x + 9 = 396
⇒ 9x = 387
⇒ x = 387/9
∴ x = 43

∴ The largest number = 3x + 6 = 3 × 43 + 6 = 135

১,৪৯০.

If n is a positive integer, then n(n + 1)(n + 2) is-

ক
even only when n is even
খ
even only when n is odd
গ
odd whenever n is odd
ঘ
divisible by 3 only when n is odd
ঙ
divisible by 4 whenever n is even

ব্যাখ্যা

Question: If n is a positive integer, then n(n + 1)(n + 2) is-

Solution:
There's a nice rule says:
The product of k consecutive integers is divisible by k, k - 1, k - 2, ..., 2, and 1
So, for example, the product of any 5 consecutive integers will be divisible by 5, 4, 3, 2 and 1
Likewise, the product of any 11 consecutive integers will be divisible by 11, 10, 9, . . . 3, 2 and 1
NOTE: the product may be divisible by other numbers as well, but these divisors are guaranteed.

Notice that n, n + 1, and n + 2 are three consecutive integers.
This means the product of n, n + 1, and n + 2 will be divisible by 3, 2 and 1

Since n(n + 1)(n + 2) is divisible by 2, then the product is ALWAYS EVEN.
This means we can eliminate answer choices A and B, because they put restrictions on when the product is even.
We can also eliminate C because it suggests that the product can be odd.

Likewise, since n(n + 1)(n + 2) is ALWAYS divisible by 3, we can eliminate answer choice D, because it puts a restriction on when the product is divisible by 3

If n is even ⇒ even × odd × even (Prod of two even numbers always divisible by 2 × 2 = 4)

১,৪৯১.

Six consecutive whole numbers are given. The sum of the first three numbers is 27. What is the sum of the last three numbers?

ক
32
খ
34
গ
36
ঘ
60

ব্যাখ্যা

Question: Six consecutive whole numbers are given. The sum of the first three numbers is 27. What is the sum of the last three numbers?

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

x - 2 + x - 1 + x = 27
⇒ 3x - 3 = 27
⇒ 3x = 27 + 3 = 30
⇒ x = 30/3
∴ x = 10

the sum of the last three numbers is = x + 1 + x + 2 + x + 3
= 10 + 1 + 10 + 2 + 10 + 3
= 36

১,৪৯২.

The 5th and 12th term of an arithmetic progression are -21 and 21 respectively. What is the 21th term?

ক
145
খ
165
গ
135
ঘ
75

ব্যাখ্যা

Let first term be 'a' and common difference be 'd'
5th term = a + 4d = - 21
12th term = a + 11d = 21
Therefore, d = 6 and a = - 45
21th term = - 45 + 20 × 6 = - 45 + 120 = 75

১,৪৯৩.

When a certain number is divided by 7, the remainder is 0, if the remainder is not 0 when the number is divided by 14, then the remainder must be -

ক
ক) 7
খ
খ) 5
গ
গ) 3
ঘ
ঘ) 8

ব্যাখ্যা

Numbers can be divided by 7 are: 7, 14, 21, 28, 35
Among them, 14, 28 are divided by 14 with no remainder
But when 7, 21, 35 these numbers are divided by 14, the remainder is 7

১,৪৯৪.

The product of three consecutive even integers is 480. Find their sum.

ক
24
খ
14
গ
28
ঘ
32

ব্যাখ্যা

Question: The product of three consecutive even integers is 480. Find their sum.

Solution:
Let the three consecutive even integers are,
x - 2, x, x + 2

Now their product is,
⇒ (x - 2)(x)(x + 2) = 480
⇒ x(x² - 4) = 480
⇒ x³ - 4x = 480

Now try even integer values,
x = 6 than (6)³ - (4 × 6) = 216 - 24 = 192 not valid
x = 8 than (8)3 - (4 × 8) = 512 - 32 = 480 is valid
So,
x = 8
Then the three even integers are,
x - 2 = 6, x = 8, x + 2 = 10

Now their sum is = 6 + 8 + 10 = 24

১,৪৯৫.

Find the smallest fraction from the following:

ক
7/20
খ
9/50
গ
3/25
ঘ
7/125

ব্যাখ্যা

7/20 = 35/100
9/50 = 18/100
3/25 = 12/100
7/125 = 56/1000 = 5.6/100

১,৪৯৬.

In each expression below, N represents a negative integer. Which expression could have a negative value?

ক
ক) N²
খ
খ) 6 - N
গ
গ) -N
ঘ
ঘ) 6 + N

ব্যাখ্যা

যেহেতু,
N ঋনাত্মক পূর্ণসংখ্যা।
ধরি, N = -7

∴ (N)² = (-7)² = 49
6 - N = 6 - (-7) = 13
- N = -(-7) = 7
6 + N = 6 + (-7) = -1

তবে অপশন D তে N এর মান -6 বা তার কম হলে এটার মানও ধণাত্মক হতো। এজন্যই প্রশ্নে 'Could have a negative value' টার্ম ব্যবহার করা হয়েছে

∴ 6 + N এর মান ঋনাত্মক হতে পারে।

১,৪৯৭.

Of two numbers, 4 times the smaller one is less than 3 times the larger one by 5. If the sum of the number is larger than 6 times their difference by 6, find the two numbers.

ক
ক) 54, 41
খ
খ) 58, 43
গ
গ) 59, 41
ঘ
ঘ) 59, 43

ব্যাখ্যা

Let the number be x and y, such that x > y.
then, 3x - 4y = 5 ..........(i)
And (x + y) - 6 (x - y) = 6 ⇔ -5x + 7y = 6 ........(ii)
Solving (i) and (ii), we get : x = 59 and y = 43
Answer : 59, 43.

১,৪৯৮.

A boy divided the numbers 7654, 8506 and 9997 by a certain largest number and he gets the same remainder in each case. What is the common remainder?

ক
ক) 156
খ
খ) 211
গ
গ) 231
ঘ
ঘ) 199

ব্যাখ্যা

If the remainder is same in each case and remainder is not given,
HCF of the differences of the numbers is the required largest number.

9997 - 7654 = 2343
9997 - 8506 = 1491
8506 - 7654 = 852
Hence, the greatest number which divides 7654, 8506 and 9997 and leaves same remainder
= HCF of 2343, 1491, 852
= 213
Now we need to find out the common remainder.
Take any of the given numbers from 7654, 8506 and 9997, say 7654
7654/213 = 35,
remainder = 199.

১,৪৯৯.

36 is divided into two parts such that 5 times of first part added to 3 times the second part makes 142. What are both parts?

ক
ক) 16, 18
খ
খ) 15, 17
গ
গ) 17, 19
ঘ
ঘ) 20, 22

ব্যাখ্যা

Let, first number be x and second number be y
Here,
x + y = 36 .... (i)
and, 5x + 3y = 142 ...(ii)

by multiplying the first equation by 3
3x + 3y = 108
5x + 3y = 142

by subtraction the second equation from the first
(3–5)x + (3–3)y = 108–142
Or, -2x = -34
Or, x = 17

by substituting x in the first equation
17 + y = 36
Or, y = 19

১,৫০০.

If sum of two consecutive numbers is multiplied by 5, then the result is 105. What are the numbers?

ক
15, 16
খ
20, 21
গ
8, 9
ঘ
10, 11

ব্যাখ্যা

Question: If sum of two consecutive numbers is multiplied by 5, then the result is 105. What are the numbers?

Solution:
ধরি
ধারাবাহিক দুটি সংখ্যা,
x এবং x + 1

প্রশ্নমতে,
⇒ {x + (x + 1)} × 5 = 105
⇒ 2x + 1 = 105/5
⇒ 2x + 1 = 21
⇒ 2x = 21 - 1
⇒ x = 20/2
∴ x = 10

∴ সংখ্যাগুলো, x = 10 এবং x + 1 = 11