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# If p is an integer, then p is divisible by how many positive integers?

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Math Expert
Joined: 02 Sep 2009
Posts: 51185
If p is an integer, then p is divisible by how many positive integers?  [#permalink]

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10 Dec 2017, 00:16
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Difficulty:

25% (medium)

Question Stats:

80% (01:04) correct 20% (01:26) wrong based on 40 sessions

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If p is an integer, then p is divisible by how many positive integers?

(1) p = 2^x, where x is a prime number.
(2) p = x^2, where x is a prime number.

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Joined: 24 Nov 2016
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Re: If p is an integer, then p is divisible by how many positive integers?  [#permalink]

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10 Dec 2017, 04:37
Bunuel wrote:
If p is an integer, then p is divisible by how many positive integers?

(1) p = 2^x, where x is a prime number.
(2) p = x^2, where x is a prime number.

(1) p = 2^x, where x is a prime number.

If x = 2 then $$2^(2) = 4$$ and p is divisible by {1,2,4} = 3 positive integers;
If x = 3 then $$2^(3) = 8$$ and p is divisible by {1,2,4,8} = 4 positive integers; not suf.

(2) p = x^2, where x is a prime number.

If x = 2 then $$(2)^2 = 4$$ and p is divisible by {1,2,4} = 3 positive integers;
If x = 3 then $$(3)^2 = 9$$ and p is divisible by {1,3,9} = 3 positive integers; suf.

The only positive integers with exactly three factors are the squares of primes.

Re: If p is an integer, then p is divisible by how many positive integers? &nbs [#permalink] 10 Dec 2017, 04:37
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