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If p and q are composite integers, is p/q an integer?

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If p and q are composite integers, is p/q an integer?  [#permalink]

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New post 14 Apr 2019, 10:35
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If p and q are composite integers, is p/q an integer?

(1) Each prime factor of q is also a factor of p.
(2) The total number of factors of q is 12, and that of p is 24
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Re: If p and q are composite integers, is p/q an integer?  [#permalink]

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New post 14 Apr 2019, 17:30
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if P=3 and Q=9, every prime factor of q is also a factor of p, but p/q is not an integer; if we invert p and q, p/q is. Therefore 1 is not sufficient.
If every factor of q is a factor of p, p/q is an integer; however, if they differ by a single prime, it is not. therefore 2 is not sufficient
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If p and q are composite integers, is p/q an integer?  [#permalink]

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New post 15 Apr 2019, 09:18
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composite numbers are non-prime numbers

(1) Each prime factor of q is also a factor of p.

let a and b prime factors for q and p.
if \(p = a^5b^3\) and \(q = a^3b^2\) then \(\frac{p}{q}\) is integer
if \(p = a^3b^2\) and \(q = a^5b^3\) then \(\frac{p}{q}\) is non-integer
insufficient ---> the trick here is that the statement does't say that "each factor of q is a factor of p"

(2) The total number of factors of q is 12, and that of p is 24

let a,b and c prime factors
if \(p = a^5b^3\) [no. of factors = (5+1)(3+1)=24] and \(q = a^3b^2\) [no. of factors =(3+1)(2+1)=12] then \(\frac{p}{q}\) is integer
if \(p = a^2b^3c^1\) [no. of factors = (2+1)(3+1)(1+1)=24] and \(q = a^3b^2\) [no. of factors = (3+1)(2+1)=12] then \(\frac{p}{q}\) is non-integer
insufficient

combining 1 & 2:
the same assumptions stated to test statement (2) shows that combining is still insufficient. E
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Re: If p and q are composite integers, is p/q an integer?  [#permalink]

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New post 24 Apr 2019, 15:49
Shouldn’t this question say “distinct prime factors”? If you just say prime factors I don’t understand why 1) is insufficient.

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Re: If p and q are composite integers, is p/q an integer?  [#permalink]

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New post 24 Apr 2019, 20:18
JDF wrote:
Shouldn’t this question say “distinct prime factors”? If you just say prime factors I don’t understand why 1) is insufficient.


hi JDF
This is the main trick.
when a question mention the prime factors, it must be talking about the distinct primes.
actually, the word "distinct" is redundant.

Think about it,
what is the distinct prime factors of 64? 2 .. correct.
what is the prime factors of 64? there is still no answer other than 2 (the others factors are not primes)

Back to statement 1:
12 & 6 has the same prime factors (which are 2 &3).
if p = 12 & q = 6, then p/q is integer
if p = 6 & q = 12, then p/q is not integer ---> so insufficient.
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Re: If p and q are composite integers, is p/q an integer?  [#permalink]

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New post 24 Apr 2019, 20:34
Thank you, that is helpful. This may be a linguistic tic on the GMAT that I need to remember. I had always learned that if the factorization is 2^3 3 the the factors are 2, 2, 2, and 3 and the distinct factors are 2 and 3. Slight difference in how things are expressed from what I’m used to.

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Re: If p and q are composite integers, is p/q an integer?   [#permalink] 24 Apr 2019, 20:34
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