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# If p and q are both positive integers such that p/9 + q/10 is also an

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Manager
Joined: 18 Jan 2017
Posts: 128
If p and q are both positive integers such that p/9 + q/10 is also an  [#permalink]

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26 Jul 2017, 03:38
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Difficulty:

5% (low)

Question Stats:

89% (01:01) correct 11% (02:38) wrong based on 88 sessions

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If p and q are both positive integers such that $$\frac{p}{9}$$+ $$\frac{q}{10}$$ is also an integer, then which one of the following numbers could p equal?

(A) 3
(B) 4
(C) 9
(D) 11
(E) 19

Source: GMAT NOVA MATH BIBLE
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Senior Manager
Joined: 28 Jun 2015
Posts: 290
Concentration: Finance
GPA: 3.5
If p and q are both positive integers such that p/9 + q/10 is also an  [#permalink]

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26 Jul 2017, 04:48
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Inten21 wrote:
If p and q are both positive integers such that $$\frac{p}{9}$$+ $$\frac{q}{10}$$ is also an integer, then which one of the following numbers could p equal?

(A) 3
(B) 4
(C) 9
(D) 11
(E) 19

Source: GMAT NOVA MATH BIBLE
A KUDO = A Thank you.

Let $$k$$ be an integer such that $$\frac{p}{9} + \frac{q}{10} = k$$.

Since all three numbers $$p$$, $$q$$, and $$k$$ are integers, among the given options, the only case could be:
integer $$(\frac{p}{9})$$ + integer $$(\frac{q}{10})$$ = integer $$(k)$$

The only value that fits the condition is (C). So, Ans - C.
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##### General Discussion
Manager
Joined: 18 Jan 2017
Posts: 128
Re: If p and q are both positive integers such that p/9 + q/10 is also an  [#permalink]

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26 Jul 2017, 09:51
TimeTraveller wrote:
Inten21 wrote:
If p and q are both positive integers such that $$\frac{p}{9}$$+ $$\frac{q}{10}$$ is also an integer, then which one of the following numbers could p equal?

(A) 3
(B) 4
(C) 9
(D) 11
(E) 19

Source: GMAT NOVA MATH BIBLE
A KUDO = A Thank you.

Let $$k$$ be an integer such that $$\frac{p}{9} + \frac{q}{10} = k$$.

Since all three numbers $$p$$, $$q$$, and $$k$$ are integers, among the given options, the only case could be:
integer $$(\frac{p}{9})$$ + integer $$(\frac{q}{10})$$ = integer $$(k)$$

The only value that fits the condition is (C). So, Ans - C.

That's absolutely correct. Great work.
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Senior SC Moderator
Joined: 22 May 2016
Posts: 2651
If p and q are both positive integers such that p/9 + q/10 is also an  [#permalink]

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26 Jul 2017, 10:14
Inten21 wrote:
If p and q are both positive integers such that $$\frac{p}{9}$$+ $$\frac{q}{10}$$ is also an integer, then which one of the following numbers could p equal?

(A) 3
(B) 4
(C) 9
(D) 11
(E) 19

Source: GMAT NOVA MATH BIBLE
A KUDO = A Thank you.

Interesting question. I like NOVA, too.

I tested one answer to get a theoretical sense of the question quickly.

Test Answer A. p = 3, yields $$\frac{1}{3}$$ + $$\frac{q}{10}$$

q is an integer. There is nothing q can be to make the sum an integer. That is, $$\frac{q}{10}$$ cannot be made to equal $$\frac{2}{3}$$, e.g., such that sum = 1.

One more quick step to cement the pattern. Plug q = 10 in. Result is $$\frac{1}{3}$$ + 1 = $$\frac{4}{3}$$ = improper fraction (sum must be integer)

So this result must be avoided: fraction + integer (and vice versa) because result = improper fraction

Because the prompt and answer choices restrict us to controlling the first expression, we must guarantee that it is not a fraction.

The only way to do that is if p is a multiple of 9. (If p = 9 or multiple of 9, q = 10 or multiple of 10.)

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If p and q are both positive integers such that p/9 + q/10 is also an   [#permalink] 26 Jul 2017, 10:14
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