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

[Max@Math Revolution]

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

[Max@Math Revolution]

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.

Therefore, B is the answer.

[jpcameron17]

In order to know whether (m^n) is a perfect square, we need to know the values of m and n.

From Statement (1), we can determine the possible values of m. Since 12,3m0 is a multiple of 4, then the sum of its digits must also be divisible by 4. For example, the number 35,789 is divisible by 4 since 32 ( from 3 + 5 + 7 + 8 + 9 = 32) is divisible by 4, but 35,788 is not divisible by 4 because 31 (from 3 + 5 + 7 + 8 + 8 = 31) is not divisible by 4. The only possible values of m that satisfy this condition are m = 2 or m = 6. Since we don't know the value of n, it is impossible to determine whether m^n is a perfect square, and hence the statement alone is insufficient.

From Statement (2), we can determine the possible values of n. Since 23,4n5 is a multiple of 9, then the sum of its digits must also be divisible by 9. For example, the number 3,969 is divisible by 9 since 27 (from 3 + 9 + 6 + 9 = 27) is divisible by 9, but the number 3,968 is not divisible by 9 because 26 (from 3 + 9 + 6 + 8 = 26) is not divisible by 9. The only possible values of n that satisfy this condition are n = 4. We also know that any number raised to a positive even integer is a perfect square (e.g. 5^2 is a perfect square, as is 6^6). Therefore this statement alone is sufficient.

The answer is (B).