If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is ?72 = (2^3)(3^2)
We are told that n^2
is divisible by 72
. Because n^2
is a perfect square, all its prime factors must have even exponents. Because 72 has a prime factor 2 with an odd exponent, which is 3, 72 lacks another 2 for the whole product to be at least a square of an integer. So if you multiply 72*2, you will get the perfect square 144. Therefore sqrt(144)
= 12, which is n. The largest positive integer that will divide n is itself, so it should be 12.