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# For positive integers m and n, is m^n a perfect square?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8568
GMAT 1: 760 Q51 V42
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For positive integers m and n, is m^n a perfect square?  [#permalink]

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25 Feb 2019, 00:08
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Difficulty:

55% (hard)

Question Stats:

60% (01:54) correct 40% (02:21) wrong based on 40 sessions

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For positive integers m and n, is m^n a perfect square?

1) The five-digit integer, 12,3m0 is a multiple of 4
2) The five-digit integer, 23,4n5 is a multiple of 9

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Re: For positive integers m and n, is m^n a perfect square?  [#permalink]

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25 Feb 2019, 00:58
MathRevolution wrote:
For positive integers m and n, is m^n a perfect square?

1) The five-digit integer, 12,3m0 is a multiple of 4
2) The five-digit integer, 23,4n5 is a multiple of 9

from 1 :for a no to be multiple of 4 ; it has to be divisible twice by 4 ; so m = 2,4,6,8
; in sufficient
from 2:
234n5 ; multiple of 9
sum of digits has to be = 9
here 14+n ; n = 4
now we
know that for any integer value raised to 4 ; will always be a perfect square of a no
IMO B
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8568
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: For positive integers m and n, is m^n a perfect square?  [#permalink]

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27 Feb 2019, 04:17
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The question asks if n is an even integer or m is a perfect square. Condition 2):
“23,4n5 is a multiple of 9” is equivalent to the statement that 2 + 3 + 4 + n + 5 = n + 14 is a multiple of 9. For this to occur, we must have n = 4 and m^n = m4 = (m^2)^2 is a perfect square. Condition 2 is sufficient.

Condition 1)
“12,3m0 is a multiple of 4” is equivalent to the statement that m is an even integers, since this is what is required for 12,3m0 to be a multiple of 4.
Thus, condition 1) tells us that m = 0, 2, 4, 6 or 8. Since we don’t know the exponent n, condition 1) is not sufficient.

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Re: For positive integers m and n, is m^n a perfect square?   [#permalink] 27 Feb 2019, 04:17
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