Number System, Problems on Number

PrepBank · পাতা ১৭ / ১৮ · ১,৬০১–১,৭০০ / ১,৭৩৬

১,৬০১.

If the numbers from 1 to 24, Which are divisible by 2 are arranged in descending order, which will be at the 7^thplace from the bottom?

ক
ক) 12
খ
খ) 14
গ
গ) 16
ঘ
ঘ) 18

ব্যাখ্যা

The descending order of the number from 1 to 24 which are divisible by 2 is:
24, 22, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2
Now count the 7th place number from the right.

The number will be at the 8th place from the bottom is 14.

১,৬০২.

The difference between a two-digit number and the number obtained by reversing its digits is 54. If the ratio of the tens digit to the units digit of the number is 3 : 1, what is the difference between the sum and the difference of its digits?

ক
2
খ
3
গ
6
ঘ
8

ব্যাখ্যা

Question: he difference between a two-digit number and the number obtained by reversing its digits is 54. If the ratio of the tens digit to the units digit of the number is 3 : 1, what is the difference between the sum and the difference of its digits?

Solution:
Let the digits of the number be:

Tens digit = x
Units digit = y
So, the number = 10x + y
The reversed number = 10y + x

Given condition: The difference between the number and its reverse number is 54:
(10x + y) - (10y + x) = 54
9x - 9y = 54
⇒ x - y = 6 ............... (1)

Given ratio of digits: Tens : Units = 3 : 1
x/y = 3/1
⇒ x = 3y.................(2)

Solve the two equations:

Substitute x = 3y into x - y = 6:
3y - y = 6
⇒ 2y = 6
⇒ y = 3
Then x = 3y = 9

Digits found: Tens = 9, Units = 3
Sum and difference of digits:

Sum = x + y = 9 + 3 = 12
Difference = x - y = 9 - 3 = 6

Difference between sum and difference of digits = 12 - 6 = 6

১,৬০৩.

If one-half of one-sixth of three-fourths of a number is 9, what is one-third of the number?

ক
24
খ
32
গ
60
ঘ
48

ব্যাখ্যা

Question: If one-half of one-sixth of three-fourths of a number is 9, what is one-third of the number?

Solution:
let the number is P

ATQ,
⇒ P × (1/2) × (1/6) × (3/4) = 9
⇒ 3P/48 = 9
⇒ P = (9 × 16)
∴ P = 144

∴ one-third of the number is = 144/3 = 48

১,৬০৪.

Half the people on a bus get off at each stop after the first and no one gets on after the first stop. If only one person gets off at stop number 7, how many people got on at the first stop?

ক
ক) 128
খ
খ) 64
গ
গ) 16
ঘ
ঘ) 32

ব্যাখ্যা

Question: Half the people on a bus get off at each stop after the first and no one gets on after the first stop. If only one person gets off at stop number 7, how many people got on at the first stop?

Solution:
7th stop only one was left and in each preceding stop twice the nos... there were so 6 stops before 7th.
so 2⁶ = 64

১,৬০৫.

Find the least number of five digits which when divided by 40, 60 and 75, leave remainders 31, 51 and 66 respectively.

ক
ক) 10,196
খ
খ) 10,199
গ
গ) 10,197
ঘ
ঘ) 10,191

ব্যাখ্যা

Difference, 40 - 31 = 9
60 - 51 = 9
75 - 66 = 9
Difference between numbers and remainder is same in each case.
Then,
The answer = {(LCM of 40, 60, 75) - 9}

40 = 2 × 2 × 2 × 5
60 = 2 × 2 × 3 × 5
75 = 3 × 5 × 5
LCM = 2 × 2 × 2 × 5 × 5 × 3 = 600

But, the least number of 5 digits = 10000
10000/600, we get remainder as 400
Then, the answer = 10000 + (600 - 400) - 9 = 10,191

১,৬০৬.

The average of ten numbers is 25. The average of seven of these numbers is 22. What is the average of the remaining three numbers?

ক
28
খ
14
গ
22
ঘ
32

ব্যাখ্যা

Question: The average of ten numbers is 25. The average of seven of these numbers is 22. What is the average of the remaining three numbers?

Solution:
১০ টি সংখ্যার গড় = ২৫
১০ টি সংখ্যার সমষ্টি = ২৫ × ১০ = ২৫০

৭ টি সংখ্যার গড় = ২২
৭ টি সংখ্যার সমষ্টি = ২২ × ৭ = ১৫৪

∴ বাকী ৩ টি সংখ্যার সমষ্টি = ২৫০ - ১৫৪ = ৯৬
∴ ৩ টি সংখ্যার গড় = ৯৬/৩ = ৩২.

১,৬০৭.

If p and q are positive integers with pq = 36, then p/q cannot be.

ক
1/4
খ
4/9
গ
1/2
ঘ
None of these

ব্যাখ্যা

Question: If p and q are positive integers with pq = 36, then p/q cannot be.

Solution:
দেওয়া আছে
pq = 36

36 = 1 × 36
= 2 × 18
= 3 × 12
= 4 × 9
= 6 × 6

p/q = 3/12 = 1/4
p/q = 4/9
p/q = 2/18 = 1/9
p/q = 1/36
p/q = 6/6 = 1

∴ p/q = 1/2 কোনভাবেই হতে পারে না।

১,৬০৮.

In a row in the theatre the seats are numbered consecutively from T1 to T50. Sumon is sitting in seat T17 and Shihab is still sitting in seat T39. How many seats are there between them?

ক
ক) 23
খ
খ) 21
গ
গ) 22
ঘ
ঘ) 20

ব্যাখ্যা

তাদের মধ্যবর্তী সিটের সংখ্যা = (39 - 17) - 1 টি
= (22 - 1) টি
= 21 টি।

১,৬০৯.

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

Let the larger number is = a
Then, the other number is = a - 5
ATQ,
a² – (a-5)² = 65
⇒ a² – a² + 10a – 25 = 65
⇒ 10a = 65 + 25 = 90
⇒ a = 90/10 = 9

১,৬১০.

If 3/p = 6 and 3/q = 15 then p - q = ?

ক
ক) 1/3
খ
খ) 2/5
গ
গ) 3/10
ঘ
ঘ) 5/6
ঙ
ঙ) None of these

ব্যাখ্যা

Question: If 3/p = 6 and 3/q = 15 then p - q = ?

Solution:
3/p = 6
⇒ p = 3/6
= 1/2

3/q = 15
⇒ q = 3/15
= 1/5

∴ p - q
= (1/2) - (1/5)
= (5 - 2)/10
= 3/10

১,৬১১.

Twenty times a positive integer is less than its square by 96. What is the integer?

ক
ক) 14
খ
খ) 18
গ
গ) 24
ঘ
ঘ) 28

ব্যাখ্যা

Question: Twenty times a positive integer is less than its square by 96. What is the integer?

Solution:
Let the integer be x.

Then,
x² - 20x = 96
⇒ x² - 20x -96=0
⇒ (x + 4) (x - 24)=0
⇒ x = 24

১,৬১২.

If the two-third of three - fourth of a number is 34, what will be the 20% of that number?

ক
13.4
খ
13.6
গ
13.7
ঘ
14

ব্যাখ্যা

Question: If the two-third of three - fourth of a number is 34, what will be the 20% of that number?

Solution:
Let the number be X.

According to the question,
(2/3) × (3/4) × X = 34
⇒ (1/2) × X = 34
⇒ X = 68

Now, 20% of 68 is = 68 × (20/100) = 13.6

১,৬১৩.

Two-third of one-fifth of one-fourth of a number is 10. What is 30% of that number?

ক
60
খ
100
গ
270
ঘ
90

ব্যাখ্যা

Question: Two-third of one-fifth of one-fourth of a number is 10. What is 30% of that number?

Solution:
Let,
The number be x.

ATQ,
(2/3) × (1/5) × (1/4) × x = 10
⇒ (1/30) × x = 10
⇒ x = 10 × 30
∴ x = 300

30% of 300 = (30 × 300)/100 = 90

১,৬১৪.

The sum of three consecutive odd integers is 44 more than the last of the numbers. What is the middle number?

ক
21
খ
23
গ
25
ঘ
27

ব্যাখ্যা

Question: The sum of three consecutive odd integers is 44 more than the last of the numbers. What is the middle number?

Solution:
Let the odd is x
So,
2nd odd is x + 2
3rd odd is x + 4

According to the question,
Sum of the odd numbers = (x + 4) + 44
The equation,
x + ( x + 2) + (x + 4) = (x + 4) + 44
⇒ 3x + 6 = x + 48
⇒ 3x - x = 48 - 6
⇒ 2x = 42
⇒ x = 42/2
∴ x = 21
First number is 21
Second number is 23

∴ The middle number is 23

১,৬১৫.

Which of the following fraction is smaller than 7/8 and greater than 4/9?

ক
ক) 17/41
খ
খ) 5/13
গ
গ) 9/10
ঘ
ঘ) 13/21

ব্যাখ্যা

Question: Which of the following fraction is smaller than 7/8 and greater than 4/9?

Solution:
7/8 = 0.875
4/9 = 0.444

Option (ক) 17/41 = 0.415
Option (খ) 5/13 = 0.385
Option (গ) 9/10 = 0.900
Option (ঘ) 13/21 = 0.690

Here, 0.444 < 0.690 < 0.875

So, Option (ঘ) 13/21 = 0.690 is smaller than 7/8 and greater than 4/9.

১,৬১৬.

If A = , then the trace of A is-

ক
9
খ
13
গ
15
ঘ
17

ব্যাখ্যা

Question: If A = , then the trace of A is-

Solution:
দেয়া আছে,
A =

ট্রেস (Trace): Matrix- এর Trace হলো একটি বর্গাকার ম্যাট্রিক্সের প্রধান কর্ণের সব উপাদানের যোগফল।
প্রধান কর্ণ: ম্যাট্রিক্সের উপরের বামদিকের কোণ থেকে নিচের ডানদিকের কোণ পর্যন্ত যে উপাদানগুলো থাকে, সেগুলোই প্রধান কর্ণ।

সুতরাং প্রদত্ত ম্যাট্রিক্স এর ট্রেস = 2 + 5 + 8 = 15

১,৬১৭.

A and B are two positive integers such that AB = 72. Which of the following cannot be the value of A + B?

ক
38
খ
27
গ
73
ঘ
20
ঙ
22

ব্যাখ্যা

Question: A and B are two positive integers such that AB = 72. Which of the following cannot be the value of A + B?

Solution:
Factor pairs of 72:
(1, 72) → A + B = 73
(2, 36) → A + B = 38
(3, 24) → A + B = 27
(4, 18) → A + B = 22
(6, 12) → A + B = 18
(8, 9) → A + B = 17

So, possible values of A + B are: 73, 38, 27, 22, 18, 17.

Among the options, 20 is not possible.

১,৬১৮.

How many integers between 1 and 100 are divisible by 3 but not by 5?

ক
ক) 27
খ
খ) 29
গ
গ) 30
ঘ
ঘ) 31

ব্যাখ্যা

From 1 to 100 = 100/3 = 33.33 ≈ 33 integers are divisible by 3
From 1 to 100 = 100/15 = 6.67 ≈ 6 integers are divisible by both 3 and 5

So, 33 - 6 = 27 integers are divisible by 3 but not by 5

১,৬১৯.

When we reverse the digits of the number 14, the increases by 27. How many other two digit numbers increases by 27 when their digits are reversed?

ক
6
খ
5
গ
4
ঘ
7
ঙ
None

ব্যাখ্যা

Question: When we reverse the digits of the number 14, the increases by 27. How many other two digit numbers increases by 27 when their digits are reversed?

Solution:
Let the numbers = (10x + y),
When the digits of the number are reversed the number becomes (10y + x)

According to question,
(10y + x) - (10x + y) = 27
Or, 9(y - x) = 27
∴ y - x = 3

Possible numbers are = (25, 36, 47, 58, 69)
Total other two digits possible numbers are 5

১,৬২০.

How many cases do you need if you have to pack 112 pairs of shoes into cases that each hold 28 shoes?

ক
ক) 8
খ
খ) 10
গ
গ) 12
ঘ
ঘ) 14

ব্যাখ্যা

এখানে,
28 টি জুতা = 14 জোড়া জুতা
∴ আবরণ (case) প্রয়োজন হবে = 112/14 = 8 টি

১,৬২১.

The value of p for which equation x² + (p – 3)x + p = 0 has real and equal roots is-

ক
ক) 9
খ
খ) 4
গ
গ) 3
ঘ
ঘ) 0

ব্যাখ্যা

Given, x² + (p − 3)x + p = 0
Here, a = 1,b = (p − 3),c = p
Given that the roots are equal,
So, Discriminant = 0
⇒ b² − 4ac = 0
Discriminant = (p − 3)² − 4(1)(p) = 0
⇒ p² + 9 − 6p − 4p = 0
⇒ p² − 10p + 9 = 0
⇒ p² − 9p − p + 9 = 0
⇒ p(p − 9) − 1(p − 9) = 0
⇒ (p − 9)(p − 1) = 0
⇒ p − 9 = 0 or p − 1 = 0
Hence, p = 9 or p = 1

১,৬২২.

If m is an even integer and n is an integer (either odd or even), then which of the following will always be even?
i) m² + n² + n ii) (m - n)(n + 1) iii) m² - n² + 1

ক
Only I
খ
Only II
গ
Only III
ঘ
Both I and II

ব্যাখ্যা

Question: If m is an even integer and n is an integer (either odd or even), then which of the following will always be even?
i) m² + n² + n ii) (m - n)(n + 1) iii) m² - n² + 1

Solution:
এখানে
m একটি জোড় সংখ্যা তাই m = 4 ধরি,
n জোড় সংখ্যা ও হতে পারে আবার বিজোড় সংখ্যাও হতে পারে।
n জোড় হলে n = 2 এবং n বিজোড় হলে n = 3 ধরি।

i)
m এবং n উভয়ে জোড় হলে
m² + n² + n = 4² + 2² + 2 = 16 + 4 + 2 = 22 যা জোড় সংখ্যা

m জোড় এবং n বিজোড় হলে
m² + n² + n = 4² + 3² + 3 = 16 + 9 + 3 = 28 যা জোড় সংখ্যা

∴ i) সর্বদা জোড়

ii)
m এবং n উভয়ে জোড় হলে
(m - n)(n + 1) = (4 - 2)(2 + 1) = 2 × 3 = 6 যা জোড় সংখ্যা

m জোড় এবং n বিজোড় হলে
(m - n)(n + 1) = (4 - 3)(3 + 1) = 1 × 4 = 4 যা জোড় সংখ্যা

∴ ii) সর্বদা জোড়

iii)
m এবং n উভয়ে জোড় হলে
m² - n² + 1 = 4² - 2² + 1 = 16 - 4 + 1 = 13 যা বিজোড় সংখ্যা

m জোড় এবং n বিজোড় হলে
m² - n² + 1 = 4² - 3² + 1 = 16 - 9 + 1 = 8 যা জোড় সংখ্যা

∴ iii) সর্বদা জোড় নয়।

১,৬২৩.

200 ÷ 25 × 4 + 12 - 3 = ?

ক
ক) 40
খ
খ) 41
গ
গ) 42
ঘ
ঘ) 43

ব্যাখ্যা

200 ÷ 25 × 4 + 12 - 3
= 200/ 25 × 4 + 12 - 3
= 8 × 4 + 12 - 3
= 32 + 12 - 3
= 44 - 3
= 41

১,৬২৪.

Find the multiple of 11 in the following numbers.

ক
ক) 112144
খ
খ) 978626
গ
গ) 869756
ঘ
ঘ) 447355

ব্যাখ্যা

Question: Find the multiple of 11 in the following numbers.

Solution:
যদি কোনো সংখ্যার জোড় স্থানের অঙ্ক ও বিজোড় স্থানের অঙ্কের যোগফলের পার্থক্য 0 হয় অথবা 11 দ্বারা বিভাজ্য হয় তবে ঐ সংখ্যাটি 11 দ্বারা বিভাজ্য হবে।
এখানে,
a. (1 + 2 + 4) - (1 + 1 + 4) = 1
b. (9 + 8 + 2) - (7 + 6 + 6) = 0
c. (8 + 9 + 5) - (6 + 7 + 6) = 3
d. (4 + 7 + 5) - (4 + 3 + 5) = 4

∴ 978626 সংখ্যাটি 11 দ্বারা বিভাজ্য।

১,৬২৫.

Rohit is 28th from the left end of a row of 50 students and Shyam is 28th from the right end of the row. How many students are sitting between them?

ক
ক) 4
খ
খ) 5
গ
গ) 2
ঘ
ঘ) 3
ঙ
ঙ) None of these

ব্যাখ্যা

রোহিত যদি বা দিক থেকে ২৮ হয় তবে ডান দিক থেকে তার অবস্থান ২৩ এবং শ্যাম যদি ডান দিক থেকে ২৮ হয় তবে বা দিক থেকে তার অবস্থান ২৩। তাদের দুজনের মাঝে ছাত্রের অবস্থান হবে {৫০ - (২৩ + ২৩)} = ৪ জন।

১,৬২৬.

Two-ninth of half of a number is 20. Find 40% of that number.

ক
60
খ
90
গ
180
ঘ
72

ব্যাখ্যা

Question: Two-ninth of half of a number is 20. Find 40% of that number.

Solution:
Let the number be x.

Given that,
Two-ninth of half of the number is 20.
⇒ (2/9) × (1/2) × x = 20
⇒ (1/9) × x = 20
⇒ x = 20 × 9
∴ x = 180

Now,
Find 40% of that number = 40% of 180
= (40/100) × 180
= 72

So 40% of that number is 72.

১,৬২৭.

A number when multiplied by 16 increases by 540. What is the number?

ক
30
খ
36
গ
42
ঘ
46

ব্যাখ্যা

Question: A number when multiplied by 16 increases by 540. What is the number?

Solution:
Let the number is X.

As per question;
16X - X = 540
⇒15X = 540
∴ X = 36

১,৬২৮.

The H.C.F of two numbers is 24. The number which can be their L.C.M is-

ক
ক) 84
খ
খ) 128
গ
গ) 148
ঘ
ঘ) 120

ব্যাখ্যা

LCM must be divisible by HCF
Here, only 120 is divisible by 24

১,৬২৯.

2 - 2 + 2 - 2 + …….. 103 terms = ?

ক
-2
খ
2
গ
1
ঘ
-1
ঙ
0

ব্যাখ্যা

Clearly, the given series is such that the sum of any odd number of terms is 2 while that of any even number of terms is 0.
Thus, sum of 103 terms is 2.

১,৬৩০.

Which of the following fractions is equal to the decimal 0.0625?

ক
ক) 5/8
খ
খ) 3/8
গ
গ) 1/26
ঘ
ঘ) 1/18
ঙ
ঙ) 5/80

ব্যাখ্যা

5 / 80

= 0.0625

১,৬৩১.

If x² is an odd number, determine the nature of x² - x.

ক
Even
খ
Odd
গ
Prime
ঘ
A perfect square

ব্যাখ্যা

Question: If x² is an odd number, determine the nature of x² - x.

Solution:
যেহেতু x² বিজোড় তাই x ও বিজোড় হবে।

এখন,
x² - x
= x(x - 1)
= (x - 1)x
∴ (x - 1) এবং x দুইটি ক্রমিক সংখ্যা।

x বিজোড় সংখ্যা হলে (x - 1) অবশ্যই জোড় সংখ্যা হবে।
কারণ দুইটি ক্রমিক সংখ্যার মধ্যে একটি বিজোড় হলে অন্যটি জোড় হবে।

সুতরাং, x ও (x - 1) এর গুনফল = x(x - 1) = x² - x একটি জোড় সংখ্যা। [জোড় × বিজোড় = জোড়]

১,৬৩২.

If a + b + c = 6 and ab + bc + ca = 10 then the value of a³ + b³ + c³ - 3abc is?

ক
ক) 36
খ
খ) 48
গ
গ) 42
ঘ
ঘ) 40

ব্যাখ্যা

a + b + c = 6
ab + bc + ca = 10
∴ (a + b + c)²= 36
⇒ a²+ b²+ c²+ 2ab + 2bc + 2ca = 36
⇒ a²+ b²+ c²+ 2(ab + bc + ca) = 36
⇒ a²+ b²+ c²+ 2 × 10 = 36
⇒ a²+ b²+ c²= 16
As we know
a³ + b³ + c³ - 3abc/(a² + b² + c² - ab - bc - ca) = a + b + c
⇒a³ + b³ + c³ - 3abc/16 - (ab + bc + ca) = 6
⇒a³ + b³ + c³ - 3abc/(16 - 10) = 6
⇒a³ + b³ + c³ - 3abc = 6× 6
⇒a³ + b³ + c³ - 3abc = 36.

১,৬৩৩.

The sum of four consecutive two-digit odd numbers, when divided by 10, become a perfect square. Which of the following can possibly be one of these four numbers?

ক
25
খ
67
গ
41
ঘ
73

ব্যাখ্যা

Question: The sum of four consecutive two-digit odd numbers, when divided by 10, become a perfect square. Which of the following can possibly be one of these four numbers?

Solution:
Using options,
We find that four consecutive odd numbers are 37, 39, 41 and 43
The sum of these 4 numbers is 160, when divided by 10 we get 16 which is a perfect square.
Thus, 41 is one of the odd numbers

১,৬৩৪.

A number when divided by 247 leaves a remainder of 35. If the same number is divided by 19, what will be the remainder?

ক
10
খ
15
গ
16
ঘ
18
ঙ
None

ব্যাখ্যা

Question: A number when divided by 247 leaves a remainder of 35. If the same number is divided by 19, what will be the remainder?

Solution:
Let the number be x and the quotient is q.

Then,
x = 247q + 35
= (19 × 13q) + (19 × 1) + 16
= 19(13q + 1) + 16

So, the given number when divided by 19 gives 16 as remainder.

১,৬৩৫.

√(16 + 16) =?

ক
ক) 4√2
খ
খ) 8√2
গ
গ) 16√2
ঘ
ঘ) 8

ব্যাখ্যা

Question: √(16 + 16) =?

Solution:
√(16 + 16) = √32
= √(2 × 16)
= 4√2

১,৬৩৬.

6*2 is three digit number with * as a missing digit. If the number is divisible by 6, the missing digit is

ক
ক) 6
খ
খ) 3
গ
গ) 7
ঘ
ঘ) 2

ব্যাখ্যা

Divisibility of 6 = Number should be multiple of 3 and 2
6*2 it is definitely divisible by 2
To be divisible by 3 Sum of all digits of the number be divisible by 3
We just try to put value in place of *
632 = 6 + 3 + 2 = 11(not divisible by 3)
662 = 6 + 6 + 2 = 14(not divisible by 3)
672 = 6 + 7 + 2 = 15( divisible by 3)
622 = 6 + 2 + 2 = 10(not divisible by 3)
∴ * is replaced by 7

১,৬৩৭.

If a and b are whole numbers such that a^b = 81, what is the value of (a + 1)^{b - 1}?

ক
64
খ
100
গ
125
ঘ
49

ব্যাখ্যা

প্রশ্ন: If a and b are whole numbers such that a^b = 81, what is the value of (a + 1)^{b - 1}?

Solution:
We know that 81 = 3⁴
∴ a = 3 and b = 4

Now,
(a + 1)^{b - 1}
= (3 + 1)^{4 - 1}
= 4³ = 64

১,৬৩৮.

A number consist of two digits, the sum of the digits is 14. If 36 is subtracted from the number, the digits are interchanged. Find the number.

ক
ক) 86
খ
খ) 95
গ
গ) 68
ঘ
ঘ) 72

ব্যাখ্যা

ধরি
একক স্থানীয় অংক x
দশক স্থানীয় অংক y

সংখ্যাটি = x + 10y

প্রশ্নমতে
x + y = 14 ....................(1)
আবার
x + 10y - 36= 10x + y
10y - y = 10x - x + 36
9y - 9x = 36
9(y - x) = 36
y - x = 4
y = 4 + x......................(2)

(1) নং সমীকরণ হতে পাই
x + 4 + x = 14
2x + 4 = 14
2x = 14 - 4
2x = 10
x = 5

x এর মান (2) নং সমীকরণে বসিয়ে পাই
y = 5 + 4
y = 9

সংখ্যাটি = x + 10y
= 5 + 10 × 9
= 5 + 90
= 95

১,৬৩৯.

How many multiples of 7 are there between 100 and 160?

ক
ক) 8
খ
খ) 7
গ
গ) 9
ঘ
ঘ) 10

ব্যাখ্যা

Question: How many multiples of 7 are there between 100 and 160?

Solution:

100 থেকে 160 এর মধ্যে 7 এর গুণিতক আছে = 8 টি
7টি গুণিতক হলো - 105, 112 119 126, 133, 140, 147,154

১,৬৪০.

The smallest number which must be subtracted from 8112 to make it exactly divisible by 99 is :

ক
ক) 93
খ
খ) 94
গ
গ) 95
ঘ
ঘ) 96

ব্যাখ্যা

On dividing 8112 by 99, we get 93 as remainder.
So, the required number to be subtracted is 93.

১,৬৪১.

If xy < 0, which of the following must be true?
i. x + y = 0 ii. 2y - 2x < 0, iii. x² + y²> 0

ক
I only
খ
II only
গ
III only
ঘ
both II and III.

ব্যাখ্যা

Question: If xy < 0, which of the following must be true?

i. x + y = 0 ii. 2y - 2x < 0, iii. x² + y²> 0

Solution:
যেহেতু
xy < 0,
1) হয় x > 0 অথবা y < 0
2) হয় x < 0 অথবা y < 0

আমরা জানি
দুইটি সংখ্যার বর্গের সমষ্টি কখনোই ঋণাত্বক হতে পারে না।
অপশন গ) x² + y² অবশ্যই যেকোন মানের জন সত্য হবে।

১,৬৪২.

If 738M6M is divisible by 11, then the value of M is:

ক
12
খ
3
গ
6
ঘ
9
ঙ
None

ব্যাখ্যা

Question: If 738M6M is divisible by 11, then the value of M is:

Solution:
A number is divisible by 11 when the difference between the total number of odd places and the total number of even places is equal to zero or multiple of 11

Therefore, M + M + 3 = 7 + 8 + 6
⇒ 2M = 18
∴ M = 9

১,৬৪৩.

Find the greatest number that will divide 43, 91, and 183 and leave the same remainder.

ক
12
খ
8
গ
3
ঘ
4

ব্যাখ্যা

Question: Find the greatest number that will divide 43, 91, and 183 and leave the same remainder.

Solution:
the number is the H.C.F of (91 - 43), (183 - 91) and (183 - 43)
= H.C.F of 48, 92 and 140
= 4

১,৬৪৪.

If √24=4.889, the value of √(8/3) is = ?

ক
ক) 1.644
খ
খ) 1.533
গ
গ) 1.633
ঘ
ঘ) 1.666

ব্যাখ্যা

= √(8/3)
= √(8×3)/(3×3)
= √(24/3)
=4.899/3
=1.633

১,৬৪৫.

The average of a, b, c is б and a - b = 4, ab = 21; What is the value of c?

ক
6
খ
7
গ
8
ঘ
9

ব্যাখ্যা

Question: The average of a, b, c is б and a - b = 4, ab = 21; What is the value of c?

Solution:
দেওয়া আছে
(a + b + c)/3 = 6
a + b + c = 18

আবার
a - b = 4
ab = 21

আমরা জানি
(a + b)² = (a - b)² + 4ab
(a + b)² = (4)² + 4×21
(a + b)² = 16 + 84
(a + b)² = 100
(a + b)² = 10²
a + b = 10

এখন
a + b + c = 18
10 + c = 18
c = 18 - 10
c = 8

১,৬৪৬.

If x/a = 4, a/y = 6, a² = 9, and ab² = - 8, then x + 2y =?

ক
- 5
খ
- 13
গ
- 10
ঘ
15

ব্যাখ্যা

Question: If x/a = 4, a/y = 6, a² = 9, and ab² = - 8, then x + 2y =?

Solution:
Square rooting the given equation a² = 9 yields two solutions: a = 3 and a = - 3
In the equation ab² = - 8, b² is positive since the square of any nonzero number is positive.
Since ab² = - 8 is a negative number, a must be negative.
Hence, keep only negative solutions for a. Thus, we get a = - 3

Substituting this value of a in the given equation x/a = 4 yields
x/(- 3) = 4
∴ x = - 12

Substituting of a = - 3 in the given equation a/y = 6 yields
- 3/y = 6
∴ y = - 3/6 = - 1/2

Hence, x + 2y = - 12 + 2(- 1/2)
= - 12 - 1
= - 13

১,৬৪৭.

The sum of three prime numbers is 100. If one of them exceeds another by 36, then one of the numbers is-

ক
ক) 7
খ
খ) 29
গ
গ) 31
ঘ
ঘ) 41

ব্যাখ্যা

Let three prime numbers are x, y + 36 and y.

According to questions,
x + y + 36 + y = 100
x + 2y = 64 .................... (1)

Putting x= 2
2y + 2 = 64
x = 31

Thus, prime numbers are 31, 67 and 2.

১,৬৪৮.

A Teacher distributes x chocolate among 30 students but 3 students absent. For this every one get one chocolate extra. Find the value of x?

ক
180
খ
250
গ
320
ঘ
270

ব্যাখ্যা

Question: A Teacher distributes x chocolate among 30 students but 3 students absent. For this every one get one chocolate extra. Find the value of x?

Solution:
Given that,
Total students = 30
Absent students = 3
Chocolates distributed = x

Now,
Each student initially gets = x/30
Each student after 3 absent = x/27
Difference between both = 1

ATQ,
(x/27) − (x/30) = 1
⇒ x{(30 − 27)/810} = 1
⇒ x(3/810) = 1
⇒ x = 810/3
∴ x = 270

∴ The value of x = 270 chocolates

১,৬৪৯.

(? - 968) ÷ 79 × 4 = 512

ক
ক) 10185
খ
খ) 10190
গ
গ) 11080
ঘ
ঘ) 11075

ব্যাখ্যা

Question: (? - 968) ÷ 79 × 4 = 512

Solution:
ধরি,
সংখ্যাটি x
(x - 968) ÷ 79 × 4 = 512
⇒ x - 968 = (512 × 79)/4
⇒ x - 968 = 10112
⇒ x = 10112 + 968
∴ x = 11080

১,৬৫০.

A 3 digit number 4a3 is added to another 3 digit number 984 to give a 4 digit number 13b7, which is divisible by 11. Then a + b = ?

ক
15
খ
12
গ
11
ঘ
10
ঙ
.

ব্যাখ্যা

Question: A 3 digit number 4a3 is added to another 3 digit number 984 to give a 4 digit number 13b7, which is divisible by 11. Then a + b = ?

Solution:
এখানে
4a3 + 984 = 13b7
a + 8 = b

যেহেতু
13b7 সংখ্যাটি 11 দ্বারা বিভাজ্য

a + 8 = b , a = 0 হলে b = 8 সংখ্যাটি 1387 যা 11 দ্বারা বিভাজ্য নয়
a + 8 = b , a = 1 হলে b = 9 সংখ্যাটি 1397 যা 11 দ্বারা বিভাজ্য

a + b= 1 + 9 = 10

১,৬৫১.

If 35% of a certain number is 49, then find the number-

ক
69
খ
115
গ
91
ঘ
140

ব্যাখ্যা

Question: If 35% of a certain number is 49, then find the number-

Solution:
Let the number be x.
Then,
⇒ 35% of x = 49
⇒ (35/100) × x = 49
⇒ 7x/20 = 49
⇒ x = (20 × 49)/7
∴ x = 140

১,৬৫২.

The difference between a two-digit number and the number obtained by changing the positions of its digits is 36. What is the difference between the two digits of that number?

ক
3
খ
4
গ
9
ঘ
Cannot be determined.

ব্যাখ্যা

Question: The difference between a two-digit number and the number obtained by changing the positions of its digits is 36. What is the difference between the two digits of that number?

Solution:
The ten's digit is x and the unit's digit is y.

As per the question:
(10x + y) - (10y + x) = 36
⇒ 10x + y - 10y - x = 36
⇒ 9x - 9y = 36
⇒ 9(x - y) = 36
∴ x - y = 4

১,৬৫৩.

In an essay competition, a winner gets a prize of Tk 100 and a participant who does not win gets a prize of Tk 25. The total prize money distributed is Tk 3,000. Find the number of winners, if the total number of participants is 63.

ক
ক) 15
খ
খ) 17
গ
গ) 19
ঘ
ঘ) 21

ব্যাখ্যা

Question: In an essay competition, a winner gets a prize of Tk 100 and a participant who does not win gets a prize of Tk 25. The total prize money distributed is Tk 3,000. Find the number of winners, if the total number of participants is 63.

Solution:
ধরি
winners এর সংখ্যা = x জন
non-winners এর সংখ্যা = 63 - x জন

প্রশ্নমতে
25(63 - x) + 100x = 3000
⇒ 1575 - 25x + 100x = 3000
⇒ 75x = 3000 - 1575
⇒ 75x = 1425
⇒ x = 1425/75
x = 19

১,৬৫৪.

Three rectangular fields having area 60 m², 84 mm² and 108 mm² are to be divided into identical rectangular flower beds, each having length 6m. Find the breadth of each flower bed-

ক
ক) 3m
খ
খ) 5m
গ
গ) 7m
ঘ
ঘ) 9m
ঙ
ঙ) None of the above

ব্যাখ্যা

We need to divide each large field into smaller flower beds such that the area of each bed is same.

So, we find the HCF of the larger fields that gives us the area of the smaller field.

HCF (60, 84, 108) = 12

Now, this HCF is the area (in m²) of each flower bed.

Also, area of a rectangular field = Length x Breadth

=> 12 = 6 x Breadth

=> Breadth = 2m

Hence, each flower bed would be 2m wide.

১,৬৫৫.

A farmer had 17 cows. All but 9 died. How many were left alive?

ক
ক) 8
খ
খ) 9
গ
গ) 16
ঘ
ঘ) 17

ব্যাখ্যা

Answer is given in the question. All but 9 died means except 9 all others died. So there are 9 alive cows.

১,৬৫৬.

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

ক
ক) 13
খ
খ) 11
গ
গ) 8
ঘ
ঘ) 9

ব্যাখ্যা

Let the larger number is = a
Then, the other number is = a - 5
ATQ,
a² – (a - 5)² = 65
⇒ a² – a² + 10a – 25 = 65
⇒ 10a = 65 + 25 = 90
⇒ a = 90/10 = 9

১,৬৫৭.

The fraction equivalent to (4/11)% is

ক
ক) 1/150
খ
খ) 1/280
গ
গ) 1/275
ঘ
ঘ) 1/250

ব্যাখ্যা

(4/11)% = (4/11) × (1/100)
= 1/275

১,৬৫৮.

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

ক
ক) 3
খ
খ) 5
গ
গ) 17
ঘ
ঘ) 19

ব্যাখ্যা

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

So, the largest prime factor is 17

১,৬৫৯.

Evaluate √(41 - √(21 + √(19 - √(9))))

ক
ক) 6
খ
খ) 5
গ
গ) 3
ঘ
ঘ) 6.4

ব্যাখ্যা

√(41 - √(21 + √(19 - √(9))))
= √(41 - √(21 + √(19 - 3)))
= √(41 - √(21 + √(16)))
= √(41 - √(21 + 4))
= √(41 - 5)
= 6

১,৬৬০.

How many "7" will you pass on the way when you count from 1 to 100?

ক
ক) 18
খ
খ) 19
গ
গ) 20
ঘ
ঘ) 21

ব্যাখ্যা

Just count numbers with 7's: 7, 17, 27, 37, 47, 57, 67, 70, 71, 72, 73, 74, 75, 76, 77(doubled!), 78, 79, 87, 97.
The answer is 20 7's.

১,৬৬১.

If the number 5 ⋆ 2 is divisible by 6, then ⋆ = ?

ক
ক) 3
খ
খ) 6
গ
গ) 7
ঘ
ঘ) 2

ব্যাখ্যা

প্রশ্নে উল্লেখিত (⋆) এর স্থানে একমাত্র 2 বসালেই সংখ্যাটি হবে (522) যা 6 দ্বারা বিভাজ্য হয়।

১,৬৬২.

The sum of prime numbers that are greater than 60 but less than 70 is:

ক
125
খ
67
গ
63
ঘ
69
ঙ
128

ব্যাখ্যা

The prime numbers that are greater than 60 but less than 70 are 61 and 67. Their sum is = 61 + 67 = 128

১,৬৬৩.

If a, b, and c are nonzero numbers and a + b = c, which of the following is equal to 1?

ক
(a - b)/c
খ
(a - c)/b
গ
(b - c)/a
ঘ
(b - a)/c
ঙ
(c - b)/a

ব্যাখ্যা

Question: If a, b, and c are nonzero numbers and a + b = c, which of the following is equal to 1?

Solution:
From a + b = c we get,
a = c - b;
b = c - a;

Only option 5:
(c - b)/a
= a/a
= 1

১,৬৬৪.

Rihan has Tk 6 more than Mohan and Tk 9 more than Sohan. All three have Tk 54 in all. Sohan has-

ক
ক) Tk 11
খ
খ) Tk 14
গ
গ) Tk 17
ঘ
ঘ) Tk 20

ব্যাখ্যা

Question: Rihan has Tk 6 more than Mohan and Tk 9 more than Sohan. All three have Tk 54 in all. Sohan has a share of-

Solution:
Let, Mohan has Tk x
Then, Rihan has Tk (x + 6)
and Sohan has = (x + 6 - 9) = x - 3

ATQ,
x + x + 6 + x - 3 = 54
⇒ 3x = 54 - 3
⇒ 3x = 51
⇒ x = 17

So, Sohan has = 17 - 3 = 14

১,৬৬৫.

If the 5-digit number 750PQ is divisible by 3, 7 and 11, then what is the value of P + 2Q = ?

ক
21
খ
17
গ
13
ঘ
16

ব্যাখ্যা

Question: If the 5-digit number 750PQ is divisible by 3, 7 and 11, then what is the value of P + 2Q = ?

Solution:
Given that,
Five-digit number 750PQ is divisible by 3, 7 and 11

Now, The LCM of 3, 7, and 11 is 231.

By taking the largest 5-digit number 75099 and dividing it by 231.
If we divide 75099 by 231 we get 325 as the quotient and 24 as the remainder.

Then, the five-digit number is 75099 - 24 = 75075.

The number = 75075 and P = 7, Q = 5

Now, P + 2Q = 7 + 2 × 5
= 7 + 10
= 17

∴ The value of P + 2Q is 17.

১,৬৬৬.

How many numbers with distinct digits are possible, products of whose digits is 28?

ক
6
খ
8
গ
12
ঘ
20

ব্যাখ্যা

Question: How many numbers with distinct digits are possible, products of whose digits is 28?

Solution:
সংখ্যাটি ২ ডিজিটের হলে , সম্ভাব্য সংখ্যা ৪৭, ৭৪

সংখ্যাটি ৩ ডিজিটের হলে সম্ভাব্য সংখ্যাগুলো ১, ৪, ৭ ডিজিট দ্বারা গঠিত হবে।
এমন সংখ্যা = ৩! = ৬

মোট সম্ভাব্য সংখ্যা = ৬ + ২
= ৮ টি

১,৬৬৭.

If a, b, c, d are four consecutive natural numbers, which of the following is a perfect square number?

ক
ক) abcd
খ
খ) ab + cd
গ
গ) abcd – 1
ঘ
ঘ) abcd + 1

ব্যাখ্যা

Question: If a, b, c, d are four consecutive natural numbers, which of the following is a perfect square number?
Solution:
আমরা জানি,
যে কোনাে চারটি ক্রমিক স্বাভাবিক সংখ্যার গুণফলের সাথে 1 যােগ করলে যােগফল একটি পূর্ণবর্গ সংখ্যা হবে।
a, b, c, d চারটি ক্রমিক স্বাভাবিক সংখ্যা।
a, b, c, d এর গুণফল = abcd

abcd গুণফলের সাথে 1 যােগ করলে যােগফল = abcd + 1
abcd + 1 একটি পূর্ণবর্গ সংখ্যা হবে।

১,৬৬৮.

Difference between a two-digit number and the number obtained by interchanging the two digits is 36, what is the difference between two digits?

ক
2
খ
4
গ
8
ঘ
12

ব্যাখ্যা

Let the ten-digit be x, the unit digit is y.

According to the question,
(10x + y) - (10y + x) = 36
⇒ 9x - 9y = 36
⇒ x - y = 4.

১,৬৬৯.

The next number in the sequence 3, 5, 8, 13, 21, 34, ____ is-

ক
ক) 58
খ
খ) 40
গ
গ) 55
ঘ
ঘ) 72

ব্যাখ্যা

আমরা জানি,
Fibonacci সংখ্যা = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... পরপর দুটি সংখ্যার যােগফল পরবর্তী সংখ্যার সমান।
সুতরাং, এই ধারার পরবর্তী সংখ্যা হবে = 21 + 34 = 55

১,৬৭০.

How many positive integers less than ten thousand are multiples of both eight and eighteen?

ক
70
খ
72
গ
138
ঘ
139

ব্যাখ্যা

Question: How many positive integers less than ten thousand are multiples of both eight and eighteen?

Solution:
৪ ও 18 এর ল.সা.গু = 72
72 দ্বারা 10000 চেয়ে ছোট যতগুলো সংখ্যাকে ভাগ করা যাবে, ৪ ও 18 দ্বারাও 10000 চেয়ে ছোট যতগুলো সংখ্যাকে ভাগ করা যাবে

∴ নির্ণেয় পূর্ণ সংখ্যা =10000/72
= 138.8
≈ 138 টি

১,৬৭১.

A positive number when decreased by 4 is equal to 21 times the reciprocal of the number. The number is?

ক
ক) 3
খ
খ) 5
গ
গ) 7
ঘ
ঘ) 9

ব্যাখ্যা

Let the number be x
Then,
⇔x−4= 21/x
⇔x²−4x−21=0
⇔(x−7)(x+3)=0
⇔x=7

১,৬৭২.

A group of hikers is walking in a line. If Javed is 9^th from the front and 12^th from the back, how many hikers are there in total?

ক
20
খ
22
গ
24
ঘ
27

ব্যাখ্যা

Question: A group of hikers is walking in a line. If Javed is 9^th from the front and 12^th from the back, how many hikers are there in total?

Solution:
Javed is 9^th from the front.
Javed is 12^th from the back.

To find the total number of hikers in the line, we can use the formula:
Total number of hikers = Position of Javed from the front + Position of Javed from the back -1

Substituting the given values:
Total number of hikers = 9 + 12 -1 = 20

Thus, there are 20 hikers in total.

১,৬৭৩.

Which number is to be added to the numerator and denominator of 7/17 to form 3/5?

ক
8
খ
9
গ
7
ঘ
11

ব্যাখ্যা

Question: Which number is to be added to the numerator and denominator of 7/17 to form 3/5?

Solution:
ধরি,
সংখ্যাটি x

প্রশ্নমতে,
(7 + x)/(17 + x) = 3/5
⇒ 5(7 + x) = 3(17 + x)
⇒ 35 + 5x = 51 + 3x
⇒ 5x - 3x = 51 - 35
⇒ 2x = 16
∴ x = 8

১,৬৭৪.

If p is odd and q is even, which expression is even?

ক
p + q
খ
pq
গ
p + 2q + 1
ঘ
Both b and c

ব্যাখ্যা

Question: If p is odd and q is even, which expression is even?

Solution:
We know,
Odd + Even = Odd
And Odd × Even = Even

Now, let p = 3 and q = 4
ক) p + q = 3 + 4 = 7 ; Odd

খ) pq = 3 × 4 = 12 ; Even

গ) p + 2q + 1 = 3 + 2 × 4 + 1 = 4 + 8 = 12 ; Even

So, correct answer is Both b and c

১,৬৭৫.

The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively is-

ক
ক) 121
খ
খ) 123
গ
গ) 127
ঘ
ঘ) 235

ব্যাখ্যা

Question: The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively is

Solution:
The number will be H.C.F (1657 - 6) = 1651 and (2037 - 5) = 2032.

So H.C.F of 1651 and 2032 = 127.

১,৬৭৬.

In 87659_21 what is the least number which can be filled in blank so that the number is divisible by 11.

ক
1
খ
2
গ
3
ঘ
4

ব্যাখ্যা

Question: In 87659_21 what is the least number which can be filled in blank so that the number is divisible by 11.

Solution:
Divisibility rule of 11:
If the sum of digits at odd and even places are equal or differ by a number divisible by 11, then the number is also divisible by 11.
Let the number of blank place be x
Now,
(8 + 6 + 9 + 2) - (7 + 5 + x + 1) = 0 or 11
⇒ 25 - 13 - x = 0 or 11
⇒ 12 - x = 0 or 11

12 - x = 0
∴ x = 12

Or
12 - x = 11
⇒ x = 12 - 11
∴ x = 1

১,৬৭৭.

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

ক
ক) 299
খ
খ) 211
গ
গ) 319
ঘ
ঘ) 322

ব্যাখ্যা

The HCF of a group of numbers will be always a factor of their LCM.
HCF is the product of all common prime factors using the least power of each common prime factor.
LCM is the product of highest powers of all prime factors.
Clearly, the numbers are (23 x 13) and (23 x 14)
∴ smaller number = (23 x 13) = 299
--------------------------------------------------------------------------
দুইটি সংখ্যার গসাগু ২৩ এবং তাদের লসাগুর দুইটি উৎপাদক ১৩ ও ১৪ হলে, ছোট সংখ্যাটি কত?
একাধিক সংখ্যার গসাগু সর্বদা তাদের লসাগুর উৎপাদক হয়। সকল সাধারণ মৌলিক উৎপাদকের গুণফল হচ্ছে গসাগু।
সংখ্যা দুইটি ২৩ × ১৩ ও ২৩ × ১৪
ছোট সংখ্যাটি ২৩ × ১৩ = ২৯৯

১,৬৭৮.

The difference of the number consisting of two digits and the number formed by interchanging the digit is always divisible by-

ক
5
খ
6
গ
7
ঘ
9

ব্যাখ্যা

Question: The difference of the number consisting of two digits and the number formed by interchanging the digit is always divisible by-

Solution:
Let the ten's digit be x and the unit's digit be y.
So, the number = 10x + y
After interchanging the positions of the number's digits, the number will be = 10y + x

∴ Difference = (10x + y) - (10y + x)
= 10x + y - 10y - x
= 9x - 9y
= 9(x - y); which is divisible by 9.

১,৬৭৯.

The sum of digits of a 3-digit number is divisible by 7. Which of these numbers satisfies it?

ক
234
খ
351
গ
142
ঘ
429

ব্যাখ্যা

Question: The sum of digits of a 3-digit number is divisible by 7. Which of these numbers satisfies it?

Solution:
A number is divisible by 7 in terms of digit sum if the sum of its digits is divisible by 7.
Check each number:
234 → 2 + 3 + 4 = 9 → 9/7 = 1 remainder 2
351 → 3 + 5 + 1 = 9 → 9/7 = 1 remainder 2
142 → 1 + 4 + 2 = 7 → 7/7 = 1
429 → 4 + 2 + 9 = 15 → 15/7 = 2 remainder 1

142 satisfies it.

১,৬৮০.

ক
ক) 19
খ
খ) 17
গ
গ) 256
ঘ
ঘ) 155

ব্যাখ্যা

Question: =?

Solution:
এখানে,
√17956 = 134
√24025 = 155

= √(134 + 155)
= √289
= 17

১,৬৮১.

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

ক
40
খ
30
গ
60
ঘ
80

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. What is the smaller number?

Solution:
Let,
The numbers be x and y, x < y.
Then
x + y = 124 ................(1)

(x + 2)/y = 1/2
⇒ y = 2x + 4 ..................(2)

x + y = 124
⇒ x + 2x + 4 = 124
⇒ 3x = 120
∴ x = 40

১,৬৮২.

The Sum of three consecutive numbers is 126. Find the highest number.

ক
41
খ
42
গ
43
ঘ
44

ব্যাখ্যা

Question: The Sum of three consecutive numbers is 126. Find the highest number.

Solution:
Let,
The numbers be x, x + 1, x + 2

ATQ,

x + x + 1 + x + 2 = 126
⇒ 3x + 3 = 126
⇒ 3x = 123
∴ x = 41

∴ Highest number 41 + 2 = 43

১,৬৮৩.

The sum of three consecutive odd natural numbers each divisible by 3 is 63. What is the smallest among them?

ক
5
খ
15
গ
21
ঘ
27

ব্যাখ্যা

Question: The sum of three consecutive odd natural numbers each divisible by 3 is 63. What is the smallest among them?

Solution:
let, the numbers 3x, 3x + 6, 3x + 12

ATQ,
3x + 3x + 6 + 3x + 12 = 63
⇒ 9x + 18 = 63
⇒ 9x = 45
∴ x = 5

The smallest among them is = 3 × 5
= 15

১,৬৮৪.

What is the square root of 0.16?

ক
0.004
খ
0.04
গ
0.4
ঘ
4

ব্যাখ্যা

Question: What is the square root of 0.16?

Solution:
The square root of 0.16 = √0.16
= 0.4

১,৬৮৫.

The least number by which 294 must be multiplied to make it a perfect square is -

ক
4
খ
6
গ
5
ঘ
3

ব্যাখ্যা

Question: The least number by which 294 must be multiplied to make it a perfect square is :

Solution:
294 = 7 × 7 × 2 × 3
Here, 2 and 3 have odd exponents.
Multiplying by 2 × 3 = 6 will make 294 a perfect square.

∴ Multiplying by 6 will make 294 a perfect square.

১,৬৮৬.

Find the value of 17² - 4².

ক
272
খ
275
গ
271
ঘ
273

ব্যাখ্যা

Question: Find the value of 17² - 4².

Solution:
17² - 4²
= (17 + 4)(17 - 4)
= 21 × 13
= 273

১,৬৮৭.

The sum of four consecutive even integers is 60. What is the value of the lowest even number?

ক
ক) 12
খ
খ) 16
গ
গ) 10
ঘ
ঘ) 14

ব্যাখ্যা

Question: The sum of four consecutive even integers is 60. What is the value of the lowest even number?

Solution:

ধরি,
১ম জোড় সংখ্যা x
২য় জোড় সংখ্যা (x + 2)
৩য় জোড় সংখ্যা (x + 4)
৪র্থ জোড় সংখ্যা (x + 6)
প্রশ্নমতে,
x + x + 2 + x + 4 + x + 6 = 60
4x + 12 = 60
4x = 60 - 12
4x = 48
x = 12

সুতরাং, ক্ষুদ্রতম জোড় সংখ্যা 12

১,৬৮৮.

The number of two digit prime numbers which remain prime even inverting the position of its digits is:

ক
4
খ
5
গ
9
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.

১,৬৮৯.

A wholesale tea dealer has 408 kilograms, 468 kilograms and 516 kilograms of three different qualities of tea. He wants it all to be packed into boxes of equal size without mixing. Find the capacity of the largest possible box.

ক
12
খ
24
গ
36
ঘ
50

ব্যাখ্যা

Question: A wholesale tea dealer has 408 kilograms, 468 kilograms and 516 kilograms of three different qualities of tea. He wants it all to be packed into boxes of equal size without mixing. Find the capacity of the largest possible box.

Solution:
The capacity of the box is H.C.F. of 408, 468 and 516.

H.C.F. of 408, 468 and 516 = 12

The capacity of the largest possible box is 12 kilograms

১,৬৯০.

If x is an even number, what is the difference between the smallest even number grater than (5x + 4) and the largest even number less than (3x + 9)?

ক
ক) 2x + 2
খ
খ) 2x - 2
গ
গ) 2x
ঘ
ঘ) None

ব্যাখ্যা

Question: If x is an even number, what is the difference between the smallest even number grater than (5x + 4) and the largest even number less than (3x + 9)?

Solution:

এখানে x জোড় সংখ্যা
(5x + 4) ও জোড় সংখ্যা হবে।
(5x + 4) এর বড় ক্ষুদ্রতম জোড় সংখ্যাটি হবে (5x + 4) এর চেয়ে 2 বেশি।অর্থাৎ (5x + 4) + 2 = 5x + 6

(3x + 9) সংখ্যাটি বিজোড় সংখ্যা।
(3x + 9) এর চেয়ে ছোট জোড় সংখ্যাটি হবে (3x + 9) এর চেয়ে 1কম অর্থাৎ (3x + 9) - 1 = 3x + 8

সংখ্যাদ্বয়ের পার্থক্য = (5x + 6) - (3x + 8)
= 5x + 6 - 3x - 8
= 2x - 2

১,৬৯১.

If a number is decreased by 4 and divided by 6, the result is 8. What would be the result if 2 is subtracted from the number and then it is divided by 5?

ক
8
খ
10
গ
12
ঘ
14

ব্যাখ্যা

Question: If a number is decreased by 4 and divided by 6, the result is 8. What would be the result if 2 is subtracted from the number and then it is divided by 5?

Solution:
Let the number be x.
Then,
(x - 4)/6 = 8
⇒ x - 4 = 48
∴ x = 52

Now, for the second condition
(x - 2)/5
= (52 - 2)5
= 10

১,৬৯২.

Which of the following is odd man out?
41, 43, 47, 53, 61, 71, 73, 81

ক
47
খ
53
গ
61
ঘ
81

ব্যাখ্যা

Each of the numbers except 81 is a prime number.

১,৬৯৩.

How many perfect squares lie between 120 and 300?

ক
6
খ
7
গ
8
ঘ
9

ব্যাখ্যা

Question: How many perfect squares lie between 120 and 300?

Solution:
We know that,
(11)² = 121 (Greater than 120 but less than 300)
(17)² = 289 (Greater than 120 but less than 300)
(18)² = 324 (Greater than 120 but not less than 300)

∴ We have 7 (11 to 17) numbers between 120 and 300 which are perfect squares.
121 = (11)²144 = (12)²169 = (13)²196 = (14)²
225 = (15)²
256 = (16)²
289 = (17)²

১,৬৯৪.

Find the greatest number which divides 120, 165 and 210 exactly leaving remainders 5, 4 and 3 respectively.

ক
5
খ
7
গ
23
ঘ
None of these

ব্যাখ্যা

Question: Find the greatest number which divides 120, 165 and 210 exactly leaving remainders 5, 4 and 3 respectively

Solution:
১১৫ = ৫ × ২৩
১৬১ = ৭ × ২৩
২০৭ = ৯ × ২৩

১১৫, ১৬১, ২০৭ এর গ সা গু = ২৩

১,৬৯৫.

A student is ranked 13th from right and 8th left. How many students are there in total?

ক
ক) 18
খ
খ) 19
গ
গ) 20
ঘ
ঘ) 21
ঙ
ঙ) 22

ব্যাখ্যা

Number of students = (13 + 8) - 1 = 20

১,৬৯৬.

Which one of the following is not a prime number?

ক
31
খ
61
গ
71
ঘ
91

ব্যাখ্যা

Question: Which one of the following is not a prime number?

Solution:
91 is divisible by 7.
So, it is not a prime number.

• ১ এর চেয়ে বড় যে সকল সংখ্যাকে শুধু ১ এবং ঐ সংখ্যা ছাড়া আর কোনো সংখ্যা দ্বারা ভাগ করা যায় না, তাদেরকে মৌলিক সংখ্যা বলে।
অর্থাৎ মৌলিক সংখ্যার উৎপাদক হবে দুইটি ১ এবং শুধুমাত্র সেই সংখ্যাটি।

১,৬৯৭.

The greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm, 12 m 95 cm is:

ক
ক) 15 cm
খ
খ) 25 cm
গ
গ) 35 cm
ঘ
ঘ) 45 cm

ব্যাখ্যা

7 m = 700 cm
3 m 85 cm = 385 cm
12 m 95 cm = 1295 cm
Required length = H.C.F. of 700 cm, 385 cm and 1295 cm
= 35 cm

১,৬৯৮.

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

ক
65
খ
72
গ
76
ঘ
80

ব্যাখ্যা

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

Solution:
let one number is x
other is 184 - x

x/3 = {(184 - x)/7} + 8
⇒ (x/3) - {(184 - x)/7 = 8
⇒ {7x - (3 × 184) + 3x} /21 = 8
⇒ 10x - 552 = 168
⇒ 10x = 720
⇒ x = 720/10 = 72

Another number = 180 - 72
= 108

∴ Smaller number is 72

১,৬৯৯.

If the sum of two numbers is 34 and their H. C. F and L. C. M are 2 and 144 respectively, the sum of the reciprocals of the two numbers is-

ক
15/144
খ
11/144
গ
17/144
ঘ
13/144

ব্যাখ্যা

Question: If the sum of two numbers is 34 and their H. C. F and L. C. M are 2 and 144 respectively, the sum of the reciprocals of the two numbers is-

Solution:
Let the two numbers are, x and y then
x + y = 34
and xy = H. C. F × L. C. M = 2 × 144 = 288

Sum of their reciprocals = (1/x) + (1/y)
= (x + y)/xy
= 34/288
= 17/144

১,৭০০.

If x is divided by 7, then the remainder is 5. If 3x is divided by 7, what is the remainder?

ক
2
খ
1
গ
3
ঘ
4
ঙ
5

ব্যাখ্যা

ATQ,
X = 7 × 1 + 5 = 12
Then, 3X = 3 × 12 = 36
So, 36 ÷ 7 = 5, remainder 1