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Number System Questions with Solutions

Number System questions reward knowing a handful of divisibility shortcuts and one idea about remainders — that they behave predictably under addition and multiplication, so you never need to compute a huge number in full. Here’s the concept, four fully worked examples, and a few to try yourself.

The concept, quickly

  • Divisibility rules let you check whether a number is divisible by a small number without doing the division: by 2 if the last digit is even; by 3 if the digit sum is divisible by 3; by 4 if the last two digits form a number divisible by 4; by 5 if the last digit is 0 or 5; by 9 if the digit sum is divisible by 9; by 11 if the difference between the sum of digits in odd and even positions (counting from the right) is divisible by 11.
  • When a number N is divided by a divisor D, N = D × Q + R, where Q is the quotient and R is the remainder, with 0 ≤ R < D. This relationship is the foundation of every remainder question.
  • Remainders behave predictably under addition and multiplication: you can take the remainder of each piece first, rather than computing a huge expression in full — useful for "find the remainder of a large power" questions, where the remainders of successive powers usually repeat in a short cycle.
  • A number is a perfect square if every exponent in its prime factorization is even; it’s a perfect cube if every exponent is a multiple of 3.

Worked examples

1. Check whether 4,572 is divisible by 9.

Solution

  1. Digit sum = 4 + 5 + 7 + 2 = 18.
  2. 18 is divisible by 9, so 4,572 is divisible by 9.

Answer: Yes, divisible by 9

2. Check whether 9,251 is divisible by 11.

Solution

  1. From the right: position 1 = 1, position 2 = 5, position 3 = 2, position 4 = 9.
  2. Sum of odd positions (1st, 3rd) = 1 + 2 = 3. Sum of even positions (2nd, 4th) = 5 + 9 = 14.
  3. Difference = |3 − 14| = 11, which is divisible by 11 — so 9,251 is divisible by 11.

Answer: Yes, divisible by 11

3. Find the remainder when 7¹⁰⁰ is divided by 5.

Solution

  1. Find the pattern of 7ⁿ mod 5: 7¹ mod 5 = 2, 7² mod 5 = 4, 7³ mod 5 = 3, 7⁴ mod 5 = 1, and then it repeats — a cycle of length 4.
  2. 100 ÷ 4 leaves no remainder, so 7¹⁰⁰ lands on the 4th position in the cycle: remainder 1.

Answer: 1

4. A number, when divided by 15, leaves a remainder of 7. Find the remainder when the same number is divided by 5.

Solution

  1. The number is N = 15Q + 7 for some quotient Q. Since 15 is a multiple of 5, the term 15Q is exactly divisible by 5.
  2. So the remainder on dividing N by 5 is just 7 mod 5 = 2. (Check with Q = 1: N = 22, and 22 ÷ 5 leaves remainder 2.)

Answer: 2

Try these yourself

Work through these the same way, then check against the answer.

1. Check whether 3,684 is divisible by 4.

Answer: Yes (last two digits, 84, are divisible by 4)

2. Check whether 61,809 is divisible by 11.

Answer: Yes (odd-position sum 23 minus even-position sum 1 = 22, divisible by 11)

3. A number, when divided by 24, leaves a remainder of 11. Find the remainder when the same number is divided by 8.

Answer: 3

Where this comes up

Number System is tested directly in the Quantitative/Numerical Aptitude section of every exam Pariksha Saathi covers — SSC CGL, SSC MTS, SSC CHSL, IBPS PO, IBPS Clerk, SBI PO and SBI Clerk.

Practice more Number System questions

This page covers the concept and a handful of worked examples. Pariksha Saathi has full topic-wise practice sets for Number System (and every other Quant/Reasoning/English topic) with instant scoring and explanations — free, no account required.

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