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.)
Answer C
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