If m is a positive integer and m^2 is divisible by 48, then

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

m^2 is a positive perfect square divisible by 48, the least such perfect square is 144=48*3=12^2. Thus the least value of m is 12, which means that m in any case must be divisible by 12.

Answer: D.

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Hope it helps.

Well m^2 is divisible by 48

if we brake 48 we will get 4*4*3

now we know that m is an integer therefore, m^2 is completely divisible by 48 then to get the square number that will be divided by 48 we need to multiply 48 by 3
so that we will get 4*4*3*3 ....
square root of the above number will be 4*3 = 12
thus m = 12 and the greatest number that will divide 12 is 12 itself ...thus the answer.

M^2 is divisible by 48 so M^2 must be multiple of 48.
If the value of M is Multiples of 12 then it will satisfy the condition. If we If M is 12 or 24 or 36 then it ans is D but if M = 48 then answer should be 16.

Is the question right? Or am i missing some thing?

This explanation looks fine but why M shouldn't be 48. There is no mention in the question about M's maximum limit.

Well, the largest integer, which will divide m, has asked in the question. This means there will be no other integer that will divide m is larger than what the answer is.
Also, It is not asked us that which of the following will be the largest integer that will divide m....

Thus m can not be 48 because the largest integer that divides 48 will be 48 ...which is not in the options, hence can not be correct. 16 is also not correct because it is the largest number in the options but larger that 16 divisors are also present.

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

ok, I will try to explain this as best as possible.
the following is a tree diagram, or columns. hope it is easy to see it

m^2____m__ 2 2
____m__ 2 2 3

so, 2 * 2* 3= 12

In another words,
find out the prime factors of 48, ( 2,2,2,2,3)

, you know that M^2 is m repeated twice, so you

should distribute the the prime factors evenly

among the 2 m's, the 3 will be left over so you

should just put it out there under one of the m's.

Now, for every columns take one common number

which is 2 for the first columns, 2 for the second

column and 3 in the last column. multiplying the

prime factors ( 2*2*3) = 12...


my advice, if you google MGMAT number properties, you will find it in a PDF, just download it and read chpt 1 and 10, very helpful for these types of problems
best of luck

its mentioned indirectly by saying m has to be an integer.

if m^2 is not a perfect square then m will not be an integer.

for m to be in an integer, m^2 has to be a perfect square.

lets say m^2=48*3 = 144 (perfect square ) => m =12 (integer)

lets say m^2 = 48(not a perfect square) = > m= sqrt(48) = 2*sqrt(12) (not an integer)

I dont Get it. Factors of 48 are= 2,2,2,2 and 3 and we know that 48 is a factor of m. Therefor 48 is also a factor of m^2.
So (2^4)*3 is a factor of m and m^2. But how do you come to the solution of 12? I don't see the link between (2^4)*3 and (2^2)*(3^2). Can someone explain please?

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m^2 is a divisor of 48. M =/= divisor of 48.

Find the prime factors of 48, [2,2,2,2,3]. When you root square a number you double the amount of it's divisor, so if you root them number it should be half. So divide in half all the repeating multiples. You know there must be a 3, since 3 is a prime and no 2 integer forms a multiple of 3. Also half the 2s are gone so you have a two 2s left. 3*2*2 = 12.

Am i right if i say:
M^2 must be a multiple of 48. 48 in prime factors: 2^4 * 3^1. Thus at least the interger must have one 3 and of course a few 2´s

Greated integer which is divisible by m and an factor of m^2 would be 12...

yeah it is clear! Thanks, i wish i could explain it always like you did now.

Bunnel thanks for the explanation

Thanks alot for the help here. I have a quick doubt, the question is asking for the largest value of the largest value of m. Why are we finding the least value of m. What am I missing here.

Thanks in advance!

The question is asking for the largest values from the given options. You are supposed to find which one of the 5 options gives you the correct answer. You are confusing between finding the "maximum possible value" when no options are given and "maximum possible value" out of the given options. For our use, the least value worked . If lets say the least did not work, we could have looked for a multiple of 12 (say 24 or 26 or 48 etc.).

Thanks alot for the response. I was also able find the explanation from on of the another post from Bunnel.

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:

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