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GMAT Quantitative Reasoning

GMAT number properties practice questions

Number properties questions reward pattern recognition more than calculation. Prime factorisation, remainders and cycles of units digits explain most of them. These ten original questions are quick once the pattern is clear, so use them to build that habit.

Key ideas

  • Prime factorise first: it gives divisor counts, LCM and HCF at once.
  • Units digits of powers repeat in short cycles, so reduce the exponent mod the cycle length.
  • HCF multiplied by LCM equals the product of two numbers.
  • Trailing zeros in a factorial come from counting factors of 5.

Common traps

  • Counting 1 or the number itself incorrectly when counting divisors.
  • Forgetting that 1 is not prime.
  • Using the wrong cycle length for units digits.

Practice questions with solutions

Try each question first, then open the solution. Aim for about two minutes per question.

  1. How many positive divisors does 72 have?

    • A 10
    • B 9
    • C 14
    • D 8
    • E 12
    Show answer and solution

    Answer: E. 12

    72 = 2³ x 3². The number of divisors is (3 + 1)(2 + 1) = 12.

  2. What is the remainder when 7^100 is divided by 5?

    • A 2
    • B 3
    • C 4
    • D 0
    • E 1
    Show answer and solution

    Answer: E. 1

    7 leaves remainder 2 on division by 5. Powers of 2 mod 5 repeat 2, 4, 3, 1 every 4 steps. 100 is a multiple of 4, so the remainder is 1.

  3. What is the sum of the first 50 positive odd integers?

    • A 1,275
    • B 2,500
    • C 2,450
    • D 2,550
    • E 5,000
    Show answer and solution

    Answer: B. 2,500

    The sum of the first n odd numbers is n². So the sum is 50² = 2,500.

  4. What is the least common multiple of 12, 18 and 30?

    • A 90
    • B 180
    • C 360
    • D 120
    • E 540
    Show answer and solution

    Answer: B. 180

    12 = 2² x 3, 18 = 2 x 3², 30 = 2 x 3 x 5. Take the highest power of each prime: 2² x 3² x 5 = 180.

  5. What is the units digit of 3^47?

    • A 1
    • B 3
    • C 7
    • D 9
    • E 5
    Show answer and solution

    Answer: C. 7

    Units digits of powers of 3 repeat 3, 9, 7, 1. 47 leaves remainder 3 when divided by 4, so the units digit is the third in the cycle, which is 7.

  6. How many multiples of 6 are there from 100 to 200, inclusive?

    • A 33
    • B 16
    • C 15
    • D 17
    • E 18
    Show answer and solution

    Answer: D. 17

    The first multiple is 102 and the last is 198. The count is (198 - 102)/6 + 1 = 17.

  7. If n is a positive integer and n² is divisible by 12, what is the least possible value of n?

    • A 2
    • B 4
    • C 3
    • D 6
    • E 12
    Show answer and solution

    Answer: D. 6

    12 = 2² x 3. For n² to contain 2² x 3, n must contain 2 and 3, so n must be a multiple of 6. The least value is 6.

  8. How many trailing zeros does 30! have?

    • A 5
    • B 7
    • C 6
    • D 3
    • E 8
    Show answer and solution

    Answer: B. 7

    Each zero needs a factor of 5 (there are plenty of 2s). Count 30/5 = 6 and 30/25 = 1, so there are 7 factors of 5, and 7 zeros.

  9. Which of the following numbers is prime?

    • A 57
    • B 97
    • C 111
    • D 87
    • E 91
    Show answer and solution

    Answer: B. 97

    91 = 7 x 13, 87 = 3 x 29, 57 = 3 x 19 and 111 = 3 x 37. 97 has no divisor other than 1 and itself.

  10. The product of two positive integers is 1,200 and their highest common factor is 10. What is their least common multiple?

    • A 240
    • B 60
    • C 1,190
    • D 120
    • E 100
    Show answer and solution

    Answer: D. 120

    For two numbers, HCF x LCM = product. So LCM = 1,200 / 10 = 120. Check with 30 and 40: HCF 10, LCM 120, product 1,200.

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