Problem Solving

BTGmoderatorAT wrote:If m is a positive integer and m^2 is divisible by 48, then the largest positive integer that must divide m is?

(A) 3
(B) 6
(C) 8
(D) 12
(E) 16

Is there a strategic approach to this question? Can any experts help?

We are given that m^2/48 = integer or (m^2)/(2^4)(3^1) = integer.

However, since m^2 is a perfect square, we need to make 48, or (2^4)(3^1), a perfect square. Since all perfect squares consist of unique prime, each raised to an even exponent, the smallest perfect square that divides into m^2 is (2^4)(3^2) = 144.

Thus, m^2/144 = integer

Since m^2 is divisible by 144, we see that the largest value that must divide m is 12.

Answer: D

Hi All,

This question has a number of logic 'shortcuts' built into it, so if you were low on time you could logically guess the correct answer. Here's how you could do it... To start, you have to note that the answer choices are numbers and one of them IS the LARGEST positive integer that will divide into M.

Since M is an integer and M^2 is divisible by 48, you should be able to logically deduce that M will be divisible by both 2 (since 48 is even) and 3 (you can use the 'rule of 3' to quickly determine that fact). So we need a number that is divisible by BOTH 2 and 3. With that information alone, we can eliminate Answer choices A, C and E. The prompt asks for the LARGEST positive integer, so between the two remaining answers, and since 48 has so many 2s in it - and that the remaining answers are 6 and 12 - it's highly likely that the correct answer is 12 (which you can prove using the other approaches offered here).

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

BTGmoderatorAT wrote:If m is a positive integer and m^2 is divisible by 48, then the largest positive integer that must divide m is?

(A) 3
(B) 6
(C) 8
(D) 12
(E) 16

Is there a strategic approach to this question? Can any experts help?

We are given that m^2/48 = integer, or (m^2)/(2^4)(3^1) = integer.

However, since m^2 is a perfect square, we need to make 48, or (2^4)(3^1), a perfect square. Since all perfect squares consist of unique primes, each raised to an even exponent, the smallest perfect square that divides into m^2 is (2^4)(3^2) = 144.

Thus, m^2/144 = integer

Since m^2 is divisible by 144, we see that the largest value that must divide m is 12.

Answer: D