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If p and q are positive integers, and the remainder obtained when p is

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If p and q are positive integers, and the remainder obtained when p is  [#permalink]

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New post 17 Mar 2011, 00:27
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If p and q are positive integers, and the remainder obtained when p is divided by q is the same as the remainder obtained when q is divided by p, which of the following is a possible value of pq?

(A) 62
(B) 55
(C) 42
(D) 36
(E) 24

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Re: If p and q are positive integers, and the remainder obtained when p is  [#permalink]

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New post 17 Mar 2011, 00:37
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I am not too sure on this. I guess it is possible only when p and q are both same. If they are both same, pq must be a perfect square.

36 is a perfect square.

Ans: "D"
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Re: If p and q are positive integers, and the remainder obtained when p is  [#permalink]

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New post 17 Mar 2011, 00:56
62=2*31
55=5*11
42=2*21=6*7=3*14
24=2*12=3*8=4*6
36=2*18=3*12=4*9=6*6
hence D

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Re: If p and q are positive integers, and the remainder obtained when p is  [#permalink]

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New post 17 Mar 2011, 01:18
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Answer should be D
Two numbers giving the same remainder when divided by each other should be same- and remainder zero. 36 only perfect square.
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Re: If p and q are positive integers, and the remainder obtained when p is  [#permalink]

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New post 23 Sep 2014, 03:25
Does anyone have an algebraic way to solve this?
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Re: If p and q are positive integers, and the remainder obtained when p is  [#permalink]

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Re: If p and q are positive integers, and the remainder obtained when p is  [#permalink]

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New post 28 Aug 2018, 14:01
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gmatjon wrote:
If p and q are positive integers, and the remainder obtained when p is divided by q is the same as the remainder obtained when q is divided by p, which of the following is a possible value of pq?

(A) 62
(B) 55
(C) 42
(D) 36
(E) 24


The remainder obtained when p is divided by q is the same as the remainder obtained when q is divided by p
This information is indirectly telling us that p = q
To explain why, let's see what happens if p does NOT equal q
If that's the case, then one value must be greater than the other value.
Let's see what happens IF it were the case that p < q.

What is the remainder when p is divided by q?
Since p < q, then p divided by q equals 0 with remainder p

IMPORTANT RULE: When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0

What is the remainder when q is divided by p?
Based on the above rule, we know that the remainder must be a number such that 0 ≤ remainder < p

Hmmmmm. In our first calculation (p ÷ q), we found that the remainder = p
In our second calculation (q ÷ p), we found that 0 ≤ remainder < p
Since it's IMPOSSIBLE for the remainder to both EQUAL p and BE LESS THAN p, we can conclude that it's impossible for p to be less than q.

Using similar logic, we can see that it's also impossible for q to be less than p.

So, it MUST be the case that p = q
So, pq = p² = the square of some integer

Check the answer choices . . . only D is the square of an integer.

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Re: If p and q are positive integers, and the remainder obtained when p is   [#permalink] 28 Aug 2018, 14:01
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