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# If m and n are positive integers, is 4^m*(1/3)^n < 1?

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Re: If m and n are positive integers, is 4^m*(1/3)^n < 1? [#permalink]
Bunuel wrote:
If m and n are positive integers, is $$4^m*(\frac{1}{3})^n < 1$$?

(1) n = 2m
(2) n = 4

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Official Solution:

If $$m$$ and $$n$$ are positive integers, is $$4^m*(\frac{1}{3})^n < 1$$?

First, let's simplify the question:

Is $$4^m*\frac{1}{3^n} < 1$$?

Is $$4^m < 3^n$$?

(1) $$n = 2m$$

Substituting $$n = 2m$$ into the question, it becomes:

Is $$4^m < 3^{2m}$$?

Is $$4^m < 9^{m}$$?

Since given that $$m$$ is a positive integer, the answer to this question is YES. Sufficient.

(2) $$n = 4$$

Substituting $$n = 4$$ into the question, it becomes:

Is $$4^m < 3^4$$?

Is $$4^m < 81$$?

If $$m=1$$, the answer is YES. However, if $$m = 100$$, the answer is NO. Not sufficient.