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Quantitative Aptitude practice

Number System Questions with Answers

Number system covers the properties of numbers: divisibility, factors, HCF and LCM, remainders, unit digits and the ordering of types of numbers. Number system questions in SSC CGL, CHSL and MTS, RRB NTPC and Group D, and bank prelims are usually short, and a good method finishes each in under 40 seconds. Typical questions ask for the largest or smallest number with a given property, or a remainder. Tip: learn the divisibility rules for 2 to 12 so that you never need long division.

Formulas you need

  • Divisibility by 3: digit sum divisible by 3; by 9: digit sum divisible by 9
  • Divisibility by 4: last two digits divisible by 4; by 8: last three digits divisible by 8
  • Divisibility by 11: (sum of digits in odd places) − (sum in even places) is 0 or a multiple of 11
  • HCF × LCM = Product of the two numbers
  • HCF of fractions = HCF of numerators ÷ LCM of denominators
  • Dividend = Divisor × Quotient + Remainder
  • Smallest number leaving the same remainder r when divided by a, b, c = LCM(a, b, c) + r
  • If the gap (divisor − remainder) is the same for all divisors, the number = LCM − gap
  • Unit digits of powers repeat in a cycle of 4 (for example 7: 7, 9, 3, 1)
  • Sum of first n natural numbers = n(n + 1) ÷ 2

Worked examples

Q1. Find the largest 4-digit number divisible by 12.

  1. The largest 4-digit number is 9999
  2. 9999 ÷ 12 gives quotient 833 and remainder 3
  3. Subtract the remainder: 9999 − 3 = 9996
  4. Check: 12 × 833 = 9996

Answer: 9996

Q2. The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.

  1. HCF × LCM = product of the numbers
  2. 12 × 180 = 2160
  3. Other number = 2160 ÷ 36 = 60

Answer: 60

Q3. Find the smallest number which leaves remainders 5, 7 and 9 when divided by 6, 8 and 10 respectively.

  1. Gap for each divisor: 6 − 5 = 1, 8 − 7 = 1, 10 − 9 = 1
  2. The gap is the same (1), so number = LCM − 1
  3. LCM of 6, 8, 10 = 120
  4. Number = 120 − 1 = 119
  5. Check: 119 = 6 × 19 + 5 = 8 × 14 + 7 = 10 × 11 + 9

Answer: 119

Shortcuts that save time

  • For remainder questions, compare divisor − remainder first. A constant gap means the answer is LCM minus that gap, with no trial and error.
  • For unit digits of large powers, divide the power by 4 and use the remainder to pick from the cycle. A remainder of 0 means the last term of the cycle.
  • Test your final answer against the options using the divisibility rules. It takes five seconds and catches a wrong option before you mark it.

Where this appears

SSC CGL Tier-1Quantitative Aptitude section
SSC CHSL Tier-1Quantitative Aptitude section
SSC MTSNumerical and Mathematical Ability section
RRB NTPC CBT-1Mathematics section
Bank PO/Clerk PrelimsQuantitative Aptitude section

Number System

Practice Set
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The sum of two digits of a 2-digit number is 9. If the number obtained by interchanging the digits exceeds the original number by 27, find the number.

Quantitative Aptitude · stress-free practice · answers reveal instantly · difficulty adapts as you go

Number System questions with answers

10 sample questions from this topic with the correct option and the solution.

  1. Q1. What is the difference between 53 and 53?

    1. 4
    2. 2
    3. 3
    4. 0

    Answer: (D) 0

    Solution
    53 - 53 = 0.

  2. Q2. Let p and q be distinct positive prime numbers such that p > q and p2 − q2 = 45. What is the value of p + q?

    1. 12
    2. 13
    3. 9
    4. 10

    Answer: (C) 9

    Solution
    Factorise: $p^2-q^2=(p-q)(p+q)=45$.
    The factor pairs of 45 are $1\times45$, $3\times15$ and $5\times9$.
    $p-q=1,\ p+q=45$ gives $p=23, q=22$, but 22 is not prime.
    $p-q=3,\ p+q=15$ gives $p=9, q=6$, and neither is prime.
    $p-q=5,\ p+q=9$ gives $p=7, q=2$, both prime. Check: $49-4=45$.
    Hence $p+q=9$.

  3. Q3. What is the product of 4 and 10?

    1. 56
    2. 40
    3. 32
    4. 52

    Answer: (B) 40

    Solution
    4 x 10 = 40.

  4. Q4. Let a, b, α and β be prime numbers. n is the number of ordered pairs of a and b where 2< a < b < 50 and b- a=2. Also, α and β are the least and the greatest prime numbers less than 50. What is the value of n(α+β)?

    1. 598
    2. 294
    3. 354
    4. 493

    Answer: (B) 294

    Solution
    Twin-prime pairs with $2<a<b<50$ and $b-a=2$ are $(3,5),(5,7),(11,13),(17,19),(29,31),(41,43)$, so $n=6$.
    The least prime less than 50 is $\alpha=2$ and the greatest is $\beta=47$, so $\alpha+\beta=49$.
    Hence $n(\alpha+\beta)=6\times49=294$.

  5. Q5. What is the sum of 45 and 11?

    1. 44
    2. 50
    3. 32
    4. 56

    Answer: (D) 56

    Solution
    45 + 11 = 56.

  6. Q6. What is the sum of 68 and 17?

    1. 61
    2. 93
    3. 85
    4. 69

    Answer: (C) 85

    Solution
    68 + 17 = 85.

  7. Q7. What is the difference between 35 and 32?

    1. 7
    2. 1
    3. 3
    4. 2

    Answer: (C) 3

    Solution
    35 - 32 = 3.

  8. Q8. What is the difference between 90 and 20?

    1. 49
    2. 56
    3. 70
    4. 77

    Answer: (C) 70

    Solution
    90 - 20 = 70.

  9. Q9. What is the sum of 96 and 33?

    1. 129
    2. 103
    3. 90
    4. 155

    Answer: (A) 129

    Solution
    96 + 33 = 129.

  10. Q10. What is the difference between 96 and 74?

    1. 16
    2. 22
    3. 14
    4. 30

    Answer: (B) 22

    Solution
    96 - 74 = 22.