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Integer Properties: Practice Question of the Week #19

This week’s practice question looks deceivingly simple– try it out to test yourself before we post the answer tomorrow. Good luck!

2600 has how many positive divisors?

  1. 6
  2. 12
  3. 18
  4. 24
  5. 48
Update: Here’s the answer/explanation post!

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7 Responses to Integer Properties: Practice Question of the Week #19

  1. MN October 4, 2011 at 1:13 pm #

    There are 24 divisors; these are,
    1, 1 x 5, 1 x 13, 1 x 25, 1 x 5 x 13, 1 x 13 x 25,
    (2^1), (2^1)x 5, (2^1)x 13, (2^1)x 25, (2^1)x 5 x 13, (2^1)x 13 x 25,
    (2^2), (2^2)x 5, (2^2)x 13, (2^2)x 25, (2^2)x 5 x 13, (2^2)x 13 x 25,
    (2^3), (2^3)x 5, (2^3)x 13, (2^3)x 25, (2^3)x 5 x 13, (2^3)x 13 x 25.
    .
    Method:
    2600 = 2^3 x 5^2 x 13
    .
    Rule of thumb: Number of divisors = Product of (1 more than exponent of prime factors).
    .
    Interesting question:
    What is the smallest number with 20 divisors? I think it’s 192. Do you agree?
    .
    MN

    • Margarette Jung
      Margarette October 4, 2011 at 2:58 pm #

      Hi, MN

      Your answer to the practice question is correct! As for your question, 192 has 14 positive divisors (2^6 * 3), and the smallest number with 20 divisors is 240, since its prime factorization is 2^4 * 3 * 5.

      Best,
      Margarette

      • MN October 4, 2011 at 5:22 pm #

        Thank you, Margarette.
        .
        And yes, it’s 240 – I was wrong earlier. Thanks again.

  2. Hussain October 4, 2011 at 5:33 am #

    answer is D , 24……. 2600 = 2^3*5^2*13^1
    so divisors turned out to be 4*3*2=24

  3. Vivek Roongta October 4, 2011 at 1:48 am #

    answer is 24

    2600 = 2^3 * 5^2 * 13

    total positive divisors = (3+1)*(2+1)*(1+1)
    = 24

  4. Aashish October 3, 2011 at 10:36 pm #

    the ans is D

  5. t7t700 October 3, 2011 at 5:54 pm #

    the answer is 6


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