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# If x and y are positive integers, is x/y an integer? (1)

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If x and y are positive integers, is x/y an integer? (1) [#permalink]  09 Sep 2007, 19:34
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If x and y are positive integers, is x/y an integer?

(1) every factor of y is also a factor of x
(2) every prime factor of y is also a prime factor of x

I have OA but cannot understand the explanation, was wondering if anyone could explain in different words?

Specifically, what confuses me is OA explanation's claim that if you test x/y = 18/8, it will satisfy criteria for (2). How can this be? As I see it:

Prime factors of 18 = (2)(3)(3)
Prime factors of 8 = (2)(2)(2)

Hence ONE prime factor of y is also a prime factor of x, but it's false that EVERY prime factor of y is also a prime factor of x. I would think you'd need three 2s, i.e. (2)(2)(2), i.e. 2^3 in x for this to be the case.

Thanks!

PS - OA is A
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I agree with A..

here is why

2) every prime factor of y is also a prime factor of X

suppose..x=6, prime factors are 3 and 2

y=18, prime factors are 3 and 2...but we have two 3s and one 2

x/y is not an integer

suppose y=6 and x=18..again x/y is an integer...

1) every factor of y is also a factor of x..means x is either a multiple of y or equal to y..sufficient
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St1:
If every factor of y is also a factor of x, then x must be a multiple of y and so x/y = integer. Sufficient.

St2:
If x = 15 and it has primes 5,3 and y = 75 which has primes 5 and 3, then x/y = non-integer. However, if x = 15 and y = 5, then x/y = integer. INsufficient.

Ans A
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I believe we have answered this question many a times

2 Postive integers can have same prime factors but also other factors that are not prime

So you cannot come to a conclusion with just that infomration

A on the other hand tells you that EVER factors of X is also a factor of Y

so X/Y is definetely an integer.
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