Numbers - Solved Questions and Answers

Last Updated : 23 Sep, 2025

A number is a collection of digits in a specific order. Numbers can be created with or without digit repetition.

Numbers, questions, and answers are provided below for you to learn and practice.

Question 1: When a number is successively divided by 35, 45, and 55, we get 18, 28, and 38 as the remainder, respectively. What is the smallest such number?
Solution:

Take lcm of divisior 35, 45, 55, we get 3465
Then find common remainder:
35-18 = 17
45-28 = 17
55-38 = 17
Common remainder is 17.
Now, lcm - common remainder => 3465-17= 3448.
Thus, 3448 is the required answer.

Question 2: How many four-digit numbers are divisible by 7?

Solution:

Smallest four-digit number = 1000.
Divide: 1000 ÷ 7 ≈ 142.857
Next integer = 143, so first multiple = 143 × 7 = 1001
Largest four-digit number = 9999.
Divide: 9999 ÷ 7 ≈ 1428.428
Integer part = 1428, so last multiple = 1428 × 7 = 9996
So the sequence is: 1001, 1008, 1015, …, 9996
Sequence properties
First term a₁ = 1001
Last term a_n = 9996
Common difference d = 7
Find the number of terms using the formula: a_n= a₁+ (n − 1)d
9996 = 1001 + (n − 1)⋅7
9996 − 1001 = (n − 1)⋅7
8995 = (n − 1)⋅7
n−1 = 8995/7 = 1285
n = 1286
The number of four-digit numbers divisible by 7 is 1286

Question 3: What would be the maximum value of 'B' in the following equation :

1 2 B
+ B 4 C
+ C 6 7
--------
1035
--------
Solution:

Only the leftmost part of the number can be of two or more digits. So, we split the answer as :
1 2 B
+ B 4 C
+ C 6 7
--------
10 3 5
--------
Now, from column 1, we can easily infer that B + C = 8.
First, let us consider B + C = 18. This is the case possible if and only if B = C = 9. So, the equation would be 129 + 949 + 967 = 2045, but we need 1035 as the answer. Thus, this is not the required case.
So, B + C = 8. For maximum 'B', we put C = 0. Therefore, B = 8.
Now, to verify our answer, we put B = 8 and C = 0 in the given equation.
1 2 8
+ 8 4 0
+ 0 6 7
--------
10 3 5
--------
Therefore, our answer B = 8 is correct.

Question 4: Which of the following are prime numbers?
(i) 247
(ii) 397
(iii) 423

Solution:

(i) 16² = 256 > 247. Prime numbers less than 16 are 2, 3, 5, 7, 11, 13 and 247 is divisible by 13. Therefore, 247 is not a prime number. It is a composite number.
(ii) 20² = 400 > 397. Prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19 but 397 is not divisible by any of these. Therefore, 397 is a prime number.
(iii) 21² = 441 > 423. Prime numbers less than 21 are 2, 3, 5, 7, 11, 13, 17, 19 and 423 is divisible by 3. Therefore, 423 is not a prime number. It is a composite number.

Question 5: Find the unit's digit in the product (17)¹⁵³ x (31)⁶².
Solution:

The unit's digit of the given equation would be the same as the unit's digit of the equation 7¹⁵³ x 1⁶².
Now, we need to find a pattern in the unit's digit when we gradually increase the powers of 7. 7¹ gives 7, 7² gives 9, 7³ gives 3, 7⁴ gives 1. So, at the fourth power, we get the unit's digit as 1. Therefore, to make our work easy, we need to write the original power (153) in multiples of 4 to the closest extent possible. We multiply this power (4) by a number such that we get nearest to 153. So, 4 x 38 = 152 and 7¹⁵² also has 1 in the unit's place.
Now, (17)¹⁵³ has 7 at unit's place and (31)⁶² has 1 at the unit's place.
Therefore, the problem simply reduces to 7 x 1 = 7.
Hence, the unit's digit is 7.

Question 6: Find the unit's digit in (17)¹⁵³ + (31)⁶².
Solution:

The unit's digit of the given equation would be the same as the unit's digit of the equation 7¹⁵³ + 1⁶².
Now, we need to find a pattern in the unit's digit when we gradually increase the powers of 7. 7¹ gives 7, 7² gives 9, 7³ gives 3, 7⁴ gives 1. So, at the fourth power, we get the unit's digit as 1. Therefore, to make our work easy, we need to write the original power (153) in multiples of 4 to the closest extent possible. We multiply this power (4) by a number such that we get nearest to 153. So, 4 x 38 = 152 and 7¹⁵² also has 1 in the unit's place.
Now, (17)¹⁵³ has 7 at unit's place and (31)⁶² has 1 at the unit's place.
Therefore, the problem simply reduces to 7 + 1 = 8.
Hence, the unit's digit is 8.

Question 7: Find the total number of prime factors in the expression (14)¹¹ x (7)² x (11)³.
Solution:

(14)¹¹ x (7)² x (11)³ = (2 x 7)¹¹ x (7)² x (11)³ = (2)¹¹ x (7)¹¹ x (7)² x (11)³ = (2)¹¹ x (7)¹³ x (11)³
Therefore, total number of prime factors = 11 + 13 + 3 = 27

Question 8: Which digits should come in place of * and # such that the number 12386*# is divisible by both 8 and 5?
Solution:

Since the given number should be divisible by 5, 0 or 5 must come in place of #. But, a number ending with 5 is never divisible by 8. So, 0 will replace #.
Now, any number is divisible by 8 if its last three digits form a number divisible by 8. The number formed by the last three digits is 6*0, which becomes divisible by 8, if * is replaced by 0 or 4 or 8.
Hence, digits in place of * can be 0 or 4 or 8 respectively.

Question 9: What is the least number that must be subtracted from 9999 to make it exactly divisible by 19?
Solution:

On dividing 9999 by 19, we get 5 as remainder. Therefore, number to be subtracted = 5.

Question 10: What is the least number that must be added to 9999 to make it exactly divisible by 19?
Solution:

On dividing 9999 by 19, we get 5 as remainder. Therefore, number to be added = 19 - 5 = 14.

Question 11: A number when divided by 340 gives a remainder of 47. What would be the remainder when the same number is divided by 17?
Solution:

The number is of the form 340a + 47 = 17 * (20a) + 17 * (2) + 13 = 17 * (20a + 2) + 13.
Therefore, on dividing this number by 17, we would get 13 as the remainder.

Question 12: Find the remainder when 3²¹ is divided by 5.
Solution:

3⁴ = 81. So, the unit's digit of 3⁴ is 1.
Therefore, the unit's digit of 3²⁰ = 1 and thus, the unit's digit of 3²¹ = 1*3 = 3.
3 when divided by 5 gives 3 as the remainder.
So, the remainder when 3²¹ is divided by 5 is 3.