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

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Intern
Joined: 17 Mar 2019
Posts: 6
If p and q are composite integers, is p/q an integer?  [#permalink]

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14 Apr 2019, 10:35
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Difficulty:

75% (hard)

Question Stats:

36% (01:23) correct 64% (01:17) wrong based on 56 sessions

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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
Intern
Joined: 24 May 2017
Posts: 13
Re: If p and q are composite integers, is p/q an integer?  [#permalink]

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14 Apr 2019, 17:30
1
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
Director
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Joined: 27 Oct 2018
Posts: 684
Location: Egypt
GPA: 3.67
WE: Pharmaceuticals (Health Care)
If p and q are composite integers, is p/q an integer?  [#permalink]

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15 Apr 2019, 09:18
1
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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Manager
Joined: 20 Apr 2019
Posts: 74
Re: If p and q are composite integers, is p/q an integer?  [#permalink]

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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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Joined: 27 Oct 2018
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WE: Pharmaceuticals (Health Care)
Re: If p and q are composite integers, is p/q an integer?  [#permalink]

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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.

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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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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