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# If p and q are positive integers, is the greatest common factor of p a

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Math Expert
Joined: 02 Sep 2009
Posts: 58445
If p and q are positive integers, is the greatest common factor of p a  [#permalink]

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03 May 2016, 09:52
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Difficulty:

35% (medium)

Question Stats:

68% (01:42) correct 32% (01:52) wrong based on 67 sessions

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If p and q are positive integers, is the greatest common factor of p and q greater than 1?

(1) p^2 = pq + p

(2) p^2 = q^2 + 2q + 1

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Re: If p and q are positive integers, is the greatest common factor of p a  [#permalink]

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03 May 2016, 10:56
1
1
(1) p^2=pq+p
=> p^2 = p(q+1)
=> p = q+1
Integers p and q are consecutive integers and have greatest common factor of 1.
Sufficient
(2) p^2=q^2+2q+1
=> p^2 = (q+1)^2
=> p = q+1
Sufficient
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If p and q are positive integers, is the greatest common factor of p a  [#permalink]

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03 May 2016, 15:20
p and q are positive integers

Statement 1 : p^2=pq+p

p^2 - pq - p = 0
p(p - q - 1) = 0
Either p = 0 or (p - q - 1) = 0
Since p is a positive integer, p can't be 0
So, (p - q - 1) = 0
p = q + 1
p and q are consecutive integers which obviously wont have the greatest common factor other than 1
Sufficient

Statement 2 : p^2=q^2+2q+1

p^2 = (q+1)^2

Since both p and q are positive integers,
we can safely find the square root of the equation as
p = q +1
p and q are consecutive integers which obviously wont have the greatest common factor other than 1
Sufficient

Correct Option : D
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Re: If p and q are positive integers, is the greatest common factor of p a  [#permalink]

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14 Jan 2019, 11:19
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Re: If p and q are positive integers, is the greatest common factor of p a   [#permalink] 14 Jan 2019, 11:19
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