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# If m and n are non-negative integers, mn=?

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Math Revolution GMAT Instructor
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If m and n are non-negative integers, mn=? [#permalink]

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01 Aug 2017, 00:59
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If m and n are non-negative integers, mn=?

1) $$9^n=3^m$$
2) $$2^n=5^m$$
[Reveal] Spoiler: OA

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Re: If m and n are non-negative integers, mn=? [#permalink]

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01 Aug 2017, 01:31
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1. $$9^n=3^m$$
This is possible when the values of n and m are as follows:
a) n=1,m=2 nm=2
b) n=2,m=4, nm=8

We are not able to come up with an unique value for the expression mn(Insufficient)

2. $$2^n=5^m$$
This is possible if and only if n=m=0.

The value of nm=0 (Sufficient)(Option B)
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Kudos [?]: 671 [1], given: 17

Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4314

Kudos [?]: 3035 [0], given: 0

GPA: 3.82
Re: If m and n are non-negative integers, mn=? [#permalink]

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03 Aug 2017, 01:25
==> In the original condition, there are 2 variables (m,n) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), from con 1), you get $$9^n=(3^2)^n=3^{2n}=3^m$$, which becomes $$2n=m$$. In order for con 2) to satisfy as well, you only get m=n=0, hence it is unique and sufficient. The answer is C. However, this is an integer question, one of the key questions, so you apply CMT 4 (A: if you get C too easily, consider A or B). For con 1), the way to satisfy $$9^n=(3^2)^n=3^{2n}=3^m$$ to $$2n=m$$ is not unique and not sufficient. For con 2), from $$2^n=5^m$$, you get $$2^n=even$$ and $$5^m=odd$$, so even≠odd. Only m=n=0 satisfies this, hence it is unique and sufficient.

Therefore, the answer is B, not C.
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Re: If m and n are non-negative integers, mn=?   [#permalink] 03 Aug 2017, 01:25
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