Problem Solving

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) 7
D) 12
E) 16

RadiumBall 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) 7
D) 12
E) 16

Lets try plugging in.

m^2 = 48 * x ( x is an positive integer )

Lets calculate the multiples of 48 which are perfect squares - 48 , 96 , 144 , 192 , 240 , 288 .....
Out of above values only 144 is a square of 12, and hence 12 is the largest positive integer that divides m.


We are ruling out option E i.e. 16 as 256 is not a multiple of 48.

Answer should be D.

m^2 is divisible by 48
factor of 48 = 2^4*3
hence m*m = 2^4 * 3k or m = 2^2 *3 or 12
hence largest integer that must divide m= 12
ans D

RadiumBall 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) 7
D) 12
E) 16

m^2=48k; where k is an integer;

taking square root we have; m=4 sqrt(3k);

i.e. m must be multiple of 4 therefore option a,b,c are straight away out;

now lets see the smallest value of k for which m becomes integer is 3; therefore m must be divisible by 12,

hence D

manpsingh87 wrote:

RadiumBall 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) 7
D) 12
E) 16

m^2=48k; where k is an integer;

taking square root we have; m=4 sqrt(3k);

i.e. m must be multiple of 4 therefore option a,b,c are straight away out;

now lets see the smallest value of k for which m becomes integer is 3; therefore m must be divisible by 12,

hence D

Is there any purpose why to look for smallest value of k above ?

Correct
OA: D

6983manish wrote:

manpsingh87 wrote:

RadiumBall 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) 7
D) 12
E) 16

m^2=48k; where k is an integer;

taking square root we have; m=4 sqrt(3k);

i.e. m must be multiple of 4 therefore option a,b,c are straight away out;

now lets see the smallest value of k for which m becomes integer is 3; therefore m must be divisible by 12,

hence D

Is there any purpose why to look for smallest value of k above ?

the purpose here is to solve the question by using fundamentals, see the problem here is to find the number which divides m from the given set of options, now consider k=27; m= 4*9=36; now 36 is not given in the options..!!!

i hope that clear your doubt.!!!

manpsingh87 wrote:

6983manish wrote:

manpsingh87 wrote:

RadiumBall 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) 7
D) 12
E) 16

m^2=48k; where k is an integer;

taking square root we have; m=4 sqrt(3k);

i.e. m must be multiple of 4 therefore option a,b,c are straight away out;

now lets see the smallest value of k for which m becomes integer is 3; therefore m must be divisible by 12,

hence D

Is there any purpose why to look for smallest value of k above ?

the purpose here is to solve the question by using fundamentals, see the problem here is to find the number which divides m from the given set of options, now consider k=27; m= 4*9=36; now 36 is not given in the options..!!!

i hope that clear your doubt.!!!

Yes , that is the purpose to find the answer in the given options, but for that "lets see the smallest value of k for which m becomes integer is 3;" is not required. We need to find the largest positive integer which should lie among the answer options as well.

My doubt was just that "finding smallest " is not required, we need to find the largest multiple and should be in answer options as well.

Thanks for response.

6983manish wrote:

manpsingh87 wrote:

6983manish wrote:

manpsingh87 wrote:

RadiumBall 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) 7
D) 12
E) 16

m^2=48k; where k is an integer;

taking square root we have; m=4 sqrt(3k);

i.e. m must be multiple of 4 therefore option a,b,c are straight away out;

now lets see the smallest value of k for which m becomes integer is 3; therefore m must be divisible by 12,

hence D

Is there any purpose why to look for smallest value of k above ?

the purpose here is to solve the question by using fundamentals, see the problem here is to find the number which divides m from the given set of options, now consider k=27; m= 4*9=36; now 36 is not given in the options..!!!

i hope that clear your doubt.!!!

Yes , that is the purpose to find the answer in the given options, but for that "lets see the smallest value of k for which m becomes integer is 3;" is not required. We need to find the largest positive integer which should lie among the answer options as well.

My doubt was just that "finding smallest " is not required, we need to find the largest multiple and should be in answer options as well.

Thanks for response.

but for that "lets see the smallest value of k for which m becomes integer is 3;" is not required.
well, i didn't mention no where in my previous posts that we're need to find the "smallest value" of k,now consider your previous post

Lets calculate the multiples of 48 which are perfect squares - 48 , 96 , 144 , 192 , 240 , 288 .....

well i'm doing the same thing by taking different values of k, why do you think that its not required ???

manpsingh87 wrote:

6983manish wrote:

manpsingh87 wrote:

6983manish wrote:

manpsingh87 wrote:

RadiumBall 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) 7
D) 12
E) 16

m^2=48k; where k is an integer;

taking square root we have; m=4 sqrt(3k);

i.e. m must be multiple of 4 therefore option a,b,c are straight away out;

now lets see the smallest value of k for which m becomes integer is 3; therefore m must be divisible by 12,

hence D

Is there any purpose why to look for smallest value of k above ?

the purpose here is to solve the question by using fundamentals, see the problem here is to find the number which divides m from the given set of options, now consider k=27; m= 4*9=36; now 36 is not given in the options..!!!

i hope that clear your doubt.!!!

Yes , that is the purpose to find the answer in the given options, but for that "lets see the smallest value of k for which m becomes integer is 3;" is not required. We need to find the largest positive integer which should lie among the answer options as well.

My doubt was just that "finding smallest " is not required, we need to find the largest multiple and should be in answer options as well.

Thanks for response.

but for that "lets see the smallest value of k for which m becomes integer is 3;" is not required.
well, i didn't mention no where in my previous posts that we're need to find the "smallest value" of k,now consider your previous post

Lets calculate the multiples of 48 which are perfect squares - 48 , 96 , 144 , 192 , 240 , 288 .....

well i'm doing the same thing by taking different values of k, why do you think that its not required ???

Ok, lets not argue more on it. I must have misunderstood your first post where you wrote "now lets see the smallest value of k for which m becomes integer is 3".

No issues, we both have got the answer. Cheers buddy.

correct D it is.

This is very similar to this problem, and the answer is also 12:

https://www.beatthegmat.com/number-prope ... tml#336271

Guys,
good explanations by all. i got to learn a lot.
but if the question asked what is the largets possible integer that CAN divide n,
then the answer would be E(16) right?

rohu27 wrote:Guys,
good explanations by all. i got to learn a lot.
but if the question asked what is the largets possible integer that CAN divide n,
then the answer would be E(16) right?

Hi Rohu27

Question is "If m is a positive integer and m^2 is divisible by 48, then the largest positive integer that must divide m is "


And since we do not have 16^2 = 256 as a multiple of 48 we are left with the largest value 12 as per the answer options. Hence we mark 12 as answer.

Hope it clarifies.

Hi Manish,
I completely get what the problem is aksing for and the explanations given have been more than useful
But I was just trying to tweak the question. the Q says the largest integer which much divide m but if the question asks for a largest integer which can divide m and gives the same options then what?

we know that m=4*sqrt(3k) so k can be 48 which makes m=48 and so m is divisible by 4(m^2 would be obviously divisible by 48 then)?
do let me know if im assuming somethng here. my point is both 12 and 16 qualify for answer choices, but the value which would always divide m(for any value) will be only 12.

hope i made myself clear.

Thanks,

6983manish wrote:

rohu27 wrote:Guys,
good explanations by all. i got to learn a lot.
but if the question asked what is the largets possible integer that CAN divide n,
then the answer would be E(16) right?

Hi Rohu27

Question is "If m is a positive integer and m^2 is divisible by 48, then the largest positive integer that must divide m is "


And since we do not have 16^2 = 256 as a multiple of 48 we are left with the largest value 12 as per the answer options. Hence we mark 12 as answer.

Hope it clarifies.