If n is a positive integer and n^2...

Question

If n is a positive integer and n^2 is divisible by 72 , then the largest positive integer that must divide n is

GMAT Panda · Accepted Answer

12 - Option B Factor 72 into its prime factors 72 = 2^3 3^2 Determine the minimum exponent of 2 in n so that n^2 includes at least a factor of 2^3 If n contains 2^k, then n^2 contains 2^2k, and we need 2k 3, so k = 2 Determine the minimum exponent of 3 in n so that n^2 includes at least a factor of 3^2 If n contains 3^m, then n^2 contains 3^2m, and we need 2m 2, so m = 1 Combine the required prime factors to find the largest integer that must divide n 2^2 3^1 = 12

Explanation	Calculations	Help
$Factor 72 into its prime factors$	$72 = 2^{3} \times 3^{2}$
$Determine the minimum exponent of 2 in n so that n^{2} includes at least a factor of 2^{3}$	$If n contains 2^{k}, then n^{2} contains 2^{2 k}, and we need 2 k \geq 3, so k = 2$
$Determine the minimum exponent of 3 in n so that n^{2} includes at least a factor of 3^{2}$	$If n contains 3^{m}, then n^{2} contains 3^{2 m}, and we need 2 m \geq 2, so m = 1$
$Combine the required prime factors to find the largest integer that must divide n$	$2^{2} \times 3^{1} = 12$

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