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CAT Quant

CAT number system practice questions

The number system is the largest single block of CAT Quant and rewards pattern recognition. Remainder cycles, prime factorisation and counting divisors appear year after year, often disguised as word problems. If you learn five or six reliable techniques, such as cycles of unit digits and the factor counting formula, you can handle most of these questions in under a minute.

Key ideas

  • Prime factorise first. The number of divisors of p^a x q^b is (a + 1)(b + 1).
  • HCF x LCM = product of the two numbers.
  • Remainders of powers repeat in cycles, so reduce the exponent by the cycle length.
  • Unit digits of powers repeat every 4 (or fewer) steps.
  • Trailing zeros in n! = n/5 + n/25 + n/125 (each rounded down).

Common traps

  • Using the cycle length without checking the exponent mod cycle correctly.
  • Forgetting that a remainder cannot be negative.
  • Counting 1 and the number itself when the question excludes them.

Practice questions with solutions

Try each question first, then open the solution. Aim for about 90 seconds per question.

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

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

    Answer: D. 2

    2^3 = 8 leaves remainder 1. Since 100 = 3 x 33 + 1, 2^100 leaves the same remainder as 2^1 = 2.

  2. How many positive factors does 180 have?

    • A 12
    • B 18
    • C 16
    • D 20
    Show answer and solution

    Answer: B. 18

    180 = 2^2 x 3^2 x 5. Factors = 3 x 3 x 2 = 18.

  3. The HCF of two numbers is 8 and their LCM is 240. If one number is 48, what is the other?

    • A 32
    • B 60
    • C 80
    • D 40
    Show answer and solution

    Answer: D. 40

    Product = 8 x 240 = 1,920. Other number = 1,920 / 48 = 40.

  4. What is the unit digit of 7^123?

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

    Answer: B. 3

    Unit digits of powers of 7 cycle 7, 9, 3, 1. Since 123 = 4 x 30 + 3, the unit digit is the third in the cycle, 3.

  5. What is the sum of all two digit multiples of 6?

    • A 780
    • B 864
    • C 810
    • D 900
    Show answer and solution

    Answer: C. 810

    The multiples run from 12 to 96. The number of terms is (96 - 12)/6 + 1 = 15. Sum = 15 x (12 + 96) / 2 = 810.

  6. What is the largest three digit number divisible by both 12 and 18?

    • A 990
    • B 936
    • C 972
    • D 900
    Show answer and solution

    Answer: C. 972

    LCM(12, 18) = 36. The largest three digit multiple of 36 is 36 x 27 = 972.

  7. How many trailing zeros does 75! have?

    Type in the answer (no options).

    Show answer and solution

    Answer: 18

    75/5 = 15 and 75/25 = 3, so 18.

  8. How many integers from 1 to 200 are divisible by neither 2 nor 3?

    Type in the answer (no options).

    Show answer and solution

    Answer: 67

    Divisible by 2: 100. By 3: 66. By 6: 33. Divisible by 2 or 3 = 100 + 66 - 33 = 133. So 200 - 133 = 67.

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