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Counting divisors practice

How many factors does 360 have? You don't need to list them. Factor it into primes and multiply.

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How it works

If n = pa × qb × rc for different primes, every divisor picks an exponent of p from 0 to a, of q from 0 to b, and so on. So the number of positive divisors is

(a + 1)(b + 1)(c + 1)

Variations change which exponents are allowed. Odd divisors: the power of 2 must be 0. Even divisors: total minus odd. Perfect-square divisors: every exponent must be even. A number has an odd number of divisors exactly when it is a perfect square.

Divisor Counting practice problems with solutions

These problems come from Mathyessy's question bank. Try each one before opening the solution.

  1. How many positive divisors does 360 have?

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    Answer: 24

    360 = 2³ × 3² × 5, so it has (3 + 1)(2 + 1)(1 + 1) = 24 divisors.

  2. How many odd divisors does 3600 have?

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    Answer: 9

    3600 = 2⁴ × 3² × 5². Odd divisors use 2⁰ only, leaving (2 + 1)(2 + 1) = 9.

  3. How many divisors of 2 to the power 7 times 3 to the power 5 are perfect squares?

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    Answer: 12

    The exponent of 2 must be even and at most 7: 0, 2, 4, 6 (4 choices). The exponent of 3 must be even and at most 5: 0, 2, 4 (3 choices). 4 × 3 = 12.

  4. What is the smallest integer with exactly 12 divisors?

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    Answer: 60

    12 divisors means the exponents plus one multiply to 12: forms p¹¹, p⁵q, p³q² or p²qr. Using the smallest primes: 2048, 96, 72 and 2² × 3 × 5 = 60. The smallest is 60.

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