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# If N is a positive integer, does N have exactly three factors?

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If N is a positive integer, does N have exactly three factors? [#permalink]
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Bunuel wrote:
If N is a positive integer, does N have exactly three factors?

(1) The integer N^2 has exactly five factors

(2) Only one factor of N is a prime number

In order to have three factors exactly, N must be a square. Most numbers have an even number of factors because factors come in pairs, take 12= 1*12 = 2*6=3*4 for example. Only squares have an odd number of factors.

Statement 1:

First of all $$N^2$$ must have factors 1, N, and $$N^2$$. N cannot be prime or else we're stuck with 3 factors only. If N was a product of 2 different prime numbers, we would have N = a*b, and $$N = a^2*b^2$$ which would give (2+1)(2+1) = 9 factors. Finally, if N was a square of a prime, $$N = a^2$$ would mean the factors are 1, a, $$a^2$$, $$a^3$$, and $$N^2 = a^4$$ which is exactly 5 factors.

Thus N is a square of a prime, and N must have exactly three factors, 1, a, and N. Sufficient.

Statement 2:

If N is prime then N has only 2 factors. N could be a square of a prime which is 3 factors. Insufficient.

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Re: If N is a positive integer, does N have exactly three factors? [#permalink]
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Re: If N is a positive integer, does N have exactly three factors? [#permalink]
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