SSC Number System Questions with Solutions

Number System is the foundation of SSC Quantitative Aptitude. Questions on divisibility, unit digits, remainders, factors and perfect squares appear in CGL, CHSL, MTS and GD papers every year, and the concepts are also used in simplification, HCF-LCM and algebra. This practice paper has 30 number system questions with answers and short solutions, along with the divisibility rules and tricks you must remember.

Practice Paper at a Glance

What Is the Number System? Basic Concept in Simple Words

The number system classifies numbers into groups and studies their properties.

• Natural numbers: 1, 2, 3, ... and whole numbers: 0, 1, 2, ...
• Integers: ..., -2, -1, 0, 1, 2, ...
• Prime numbers have exactly two factors (1 and the number itself). 2 is the only even prime. Composite numbers have more than two factors.
• Rational numbers can be written as p/q (q is not zero). Irrational numbers such as the square root of 2 cannot be written as p/q.
• Co-prime numbers have only 1 as a common factor.

Divisibility Rules, Formulas and Shortcut Tricks for SSC Number System

• Sum of first n natural numbers = n(n + 1)/2. Sum of first n odd numbers = n^2. Sum of first n even numbers = n(n + 1).
• Sum of squares of the first n natural numbers = n(n + 1)(2n + 1)/6.
• Unit digit cycles: 2 (2, 4, 8, 6), 3 (3, 9, 7, 1), 7 (7, 9, 3, 1), 8 (8, 4, 2, 6) repeat every 4 powers. 4 and 9 repeat every 2 powers. 0, 1, 5 and 6 always stay the same.
• Number of factors: if N = a^p x b^q x c^r, the number of factors is (p + 1)(q + 1)(r + 1).
• Trailing zeros in n! = n/5 + n/25 + n/125 + ... (take the whole number part).
• Remainder trick: the remainder of a product is the product of remainders. (a x b) mod n = ((a mod n) x (b mod n)) mod n.
• a^2 - b^2 = (a + b)(a - b) helps in quick calculations.

Set 1: Basic Number System Questions (Q1–Q10)

Q1. Find the sum of the first 50 natural numbers.
Answer: 1,275
Trick: Sum = n(n + 1)/2 = 50 x 51 / 2 = 1,275.

Q2. How many prime numbers are there between 1 and 20?
Answer: 8
Solution: The primes are 2, 3, 5, 7, 11, 13, 17 and 19.

Q3. Find the digit in place of * so that the number 53*4 is divisible by 9.
Answer: 6
Solution: Sum of known digits = 5 + 3 + 4 = 12. To reach 18, * = 6.

Q4. What is the sum of the first 10 odd numbers?
Answer: 100
Trick: Sum of the first n odd numbers = n^2 = 100.

Q5. Find the sum of the squares of the first 5 natural numbers.
Answer: 55
Solution: 1 + 4 + 9 + 16 + 25 = 55, or 5 x 6 x 11 / 6 = 55.

Q6. Find the largest 3-digit number divisible by 11.
Answer: 990
Solution: 999 divided by 11 gives remainder 9. So 999 - 9 = 990.

Q7. How many factors does 36 have?
Answer: 9
Trick: 36 = 2^2 x 3^2. Number of factors = (2 + 1)(2 + 1) = 9.

Q8. Find the unit digit of 7^23.
Answer: 3
Trick: The cycle of 7 is 7, 9, 3, 1 (length 4). 23 divided by 4 leaves remainder 3, so the unit digit is the third value = 3.

Q9. Find the remainder when 2^10 is divided by 7.
Answer: 2
Solution: 2^3 = 8 which leaves remainder 1. 2^10 = (2^3)^3 x 2 leaves remainder 1 x 2 = 2.

Q10. Find the smallest 3-digit prime number.
Answer: 101
Solution: 100 is not prime. 101 is not divisible by 2, 3, 5 or 7, so it is prime.

Set 2: Divisibility, Remainder and Digit Questions (Q11–Q20)

Q11. Find the remainder when 5^100 is divided by 4.
Answer: 1
Trick: 5 leaves remainder 1 when divided by 4, so 5^100 leaves remainder 1^100 = 1.

Q12. A number leaves remainder 5 when divided by 7. What is the remainder when twice the number is divided by 7?
Answer: 3
Solution: Remainder = 2 x 5 = 10, and 10 divided by 7 leaves 3.

Q13. A number when divided by 56 leaves remainder 29. What is the remainder when it is divided by 8?
Answer: 5
Solution: 56 is a multiple of 8. So the remainder is 29 divided by 8, which is 5.

Q14. The sum of two numbers is 25 and their product is 144. Find the difference between them.
Answer: 7
Solution: The numbers are 9 and 16 (sum 25, product 144). Difference = 7.

Q15. The sum of the digits of a two-digit number is 9. If the digits are interchanged, the new number is 27 more than the original. Find the number.
Answer: 36
Solution: Let the number be 10a + b with a + b = 9. 9(b - a) = 27, so b - a = 3. Then a = 3 and b = 6. The number is 36.

Q16. How many zeros are there at the end of 100! (100 factorial)?
Answer: 24
Trick: 100/5 = 20 and 100/25 = 4. Total zeros = 20 + 4 = 24.

Q17. What is the smallest number that must be added to 1,000 so that it is exactly divisible by 7?
Answer: 1
Solution: 1,000 divided by 7 leaves remainder 6. We need to add 7 - 6 = 1.

Q18. Find the greatest 4-digit number exactly divisible by 88.
Answer: 9,944
Solution: 9,999 divided by 88 leaves remainder 55. So 9,999 - 55 = 9,944.

Q19. What is the sum of all even numbers from 1 to 100?
Answer: 2,550
Trick: Even numbers 2 to 100 are 50 numbers. Sum = n(n + 1) = 50 x 51 = 2,550.

Q20. By what smallest number should 12 be multiplied to get a perfect square?
Answer: 3
Solution: 12 = 2^2 x 3. Multiply by 3 to get 2^2 x 3^2 = 36.

Set 3: Mixed and Exam-Level Questions (Q21–Q30)

Q21. Find the unit digit of 234 x 456 x 123.
Answer: 2
Trick: Unit digits: 4 x 6 = 24 gives 4. Then 4 x 3 = 12 gives unit digit 2.

Q22. How many factors does 360 have?
Answer: 24
Trick: 360 = 2^3 x 3^2 x 5. Factors = (3 + 1)(2 + 1)(1 + 1) = 24.

Q23. Find the sum of all the factors of 12.
Answer: 28
Solution: Factors are 1, 2, 3, 4, 6 and 12. Sum = 28.

Q24. Find the digit * so that the number 47*32 is divisible by 9.
Answer: 2
Solution: Sum of digits = 4 + 7 + 3 + 2 = 16. To make it 18, * = 2.

Q25. Find x so that 5x38 is divisible by 11.
Answer: 0
Solution: (5 + 3) - (x + 8) = 0 - x must be 0 or a multiple of 11. So x = 0 (5,038 = 11 x 458).

Q26. The sum of three consecutive odd numbers is 57. Find the largest number.
Answer: 21
Solution: Middle number = 57 / 3 = 19. The numbers are 17, 19 and 21. Largest = 21.

Q27. What is the smallest number that must be added to 5,000 to make it a perfect square?
Answer: 41
Solution: 70^2 = 4,900 and 71^2 = 5,041. So add 5,041 - 5,000 = 41.

Q28. Find the square root of 2,025.
Answer: 45
Solution: 45 x 45 = 2,025, so the square root is 45.

Q29. Express the recurring decimal 0.272727... as a fraction in its simplest form.
Answer: 3/11
Trick: 0.2727... = 27/99 = 3/11.

Q30. Find the value of 17^2 - 13^2.
Answer: 120
Trick: Using a^2 - b^2 = (a + b)(a - b): (17 + 13)(17 - 13) = 30 x 4 = 120.

Common Mistakes in Number System

• Applying the divisibility rule of 11 incorrectly by not alternating the digit positions.
• Forgetting that 1 is neither prime nor composite.
• Not reducing the power with the cycle length in unit digit questions.
• Using n(n + 1)/2 for the sum of even numbers (use n(n + 1)).
• Counting only the prime factors and not adding 1 to each power when finding the number of factors.
• Forgetting to add the factor 25 (and 125) while counting trailing zeros.

How to Practice These 30 Questions

1. Learn the divisibility table and unit digit cycles first.
2. Solve Set 1 quickly in 6 minutes to warm up.
3. Attempt Set 2 and note the type of each question (remainder, digit or sum).
4. Set 3 is exam level, so use the shortcuts and avoid long calculation.
5. After finishing, write the formulas you forgot on a revision card.

Related Topics to Study Next

• HCF and LCM Questions
• SSC Algebra Questions
• Average Questions
• SSC Previous Year Papers

Frequently Asked Questions

What is the number system in SSC maths?

The number system is the study of types of numbers (natural, whole, integers, rational, irrational, prime and composite) and their properties such as divisibility, factors, remainders and unit digits.

What are the divisibility rules for 3, 9 and 11?

A number is divisible by 3 if its digit sum is divisible by 3 and by 9 if its digit sum is divisible by 9. For 11, the difference between the sum of digits at odd and even places must be 0 or a multiple of 11.

How do you find the unit digit of a power?

Find the cycle of the unit digit of the base, divide the power by the cycle length and use the remainder to pick the unit digit. If the remainder is 0, take the last digit of the cycle.

How do you find the number of factors of a number?

Write the number as a product of prime powers a^p x b^q x c^r. The number of factors is (p + 1)(q + 1)(r + 1).

How do you find trailing zeros in a factorial?

Divide the number by 5, 25, 125 and so on, ignore decimals and add the results. For 100!, it is 20 + 4 = 24.

What is the sum of the first n natural numbers?

The sum is n(n + 1)/2. The sum of the first n odd numbers is n^2 and the sum of the first n even numbers is n(n + 1).

How do you find the remainder of a large power?

Reduce the base using its remainder, find a power that gives remainder 1 or -1 and use that pattern. For example 2^3 leaves remainder 1 when divided by 7.

What is the difference between prime and co-prime numbers?

A prime number has only two factors, 1 and itself. Co-prime numbers are two or more numbers whose only common factor is 1, such as 8 and 15.

How many number system questions come in SSC CGL?

Usually two to three questions per shift come from number system, including divisibility, remainders, unit digit and number properties.

How should I prepare the number system for SSC exams?

Memorise the divisibility rules and unit digit cycles, practise one set daily, learn the formulas for sums and factors and revise using previous year papers.