Problem Solving

NandishSS wrote:If m is a positive integer and m² is divisible by 48, then the largest positive integer that must divide m is?
(A) 3
(B) 6
(C) 8
(D) 12
(E) 16

ASIDE-----------------------------------
A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)
--------------------------------------

m² is divisible by 48
48 = (2)(2)(2)(2)(3)
This tells us that m² = (2)(2)(2)(2)(3)(?)(?)(?)(?)(?)
NOTE: the various (?)'s includes other possible prime numbers in the prime factorization of m²

So, we know how the prime factorization of m² looks.
What does this tell us about the prime factorization of m?
First, since there are FOUR 2's in the prime factorization of m², we know that the prime factorization of m will include at least TWO 2's, since (2)(2) x (2)(2) = (2)(2)(2)(2)

Likewise, since there is ONE 3 in the prime factorization of m², we know that the prime factorization of m will include at least ONE 3.

So, we can be certain that m = (2)(2)(3)(?)(?)(?)(?)(?)
Since (2)(2)(3) = 12, we can be certain that 12 is a divisor of m.

Answer: D

NandishSS wrote:If m is a positive integer and m² is divisible by 48, then the largest positive integer that must divide m is?
(A) 3
(B) 6
(C) 8
(D) 12
(E) 16

ANOTHER APPROACH:

m² is divisible by 48
48 = (2)(2)(2)(2)(3)
Notice that 48 is NOT the square of an integer.
So, we know that m² does NOT equal 48

So, what COULD be the value of m²?
From my post above, we know that m² = (2)(2)(2)(2)(3)(?)(?)(?)(?)(?)

Also, from the above post, we know that, since there is ONE 3 in the prime factorization of m², we know that the prime factorization of m will include at least ONE 3. This also tells us that the prime factorization of m² must consist of TWO 3's.

When we add a second 3 to that prime factorization, we get, m² = (2)(2)(2)(2)(3)(3) = 144, and 144 IS the square of an integer.

So, the SMALLEST possible value of m² is 144
This means the SMALLEST possible value of m is 12
So, the biggest integer that MUST divide m is 12

Answer: D

Cheers,
Brent

NandishSS 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

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 divides m is 12.

Answer: D

Hi NandishSS,

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

NandishSS 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

OA:D

Source:GMATPrep QP1

I see many great solutions to this question.

Here is mine. I think this one is a shorter one.

We know that m^2 is divisible by 48, thus m^2 = 48K; where K is a positive number.

m^2 = 48K => m = √(48K) = √(2x2x2x2x3xK)

=> m = 4.√(3K)

Since m is an integer, √(3K) must also be an integer. The minimum value of K would be 3.

=> m = 4.√(3K) = 4.√(3x3) = 4x3 = 12

Thus, the largest positive integer that must divide m is 12.

The correct answer: D

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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