If the least common multiple of a positive integer n and 50 is 100, wh

Question

If the least common multiple of a positive integer n and 50 is 100, what is the sum of all possible values n ?

A. 4
B. 20
C. 104
D. 120
E. 124

M37-02

Bunuel · Accepted Answer

Official Solution:

If the least common multiple of a positive integer  n  and 50 is 100, what is the sum of all possible values  n  ?

A.  4 
B.  20 
C.  104 
D.  120 
E.  124

Given:  LCM(n, \ 50)=100 ;
 
 LCM(n, \ 2*5^2)=2^2*5^2 .
 
 n  must be a multiple of  2^2  (because 50 is not and the LCM is). Also, the only other prime  n  can have is 5 (because 100 does not have any other prime but 2).
 
So,  n  can be  2^2=4 ,  2^2*5=20  and  2^2*5^2=100 . 
 
The sum of all possible values  n  is therefore  4+20+100=124 .

Answer: E

BrentGMATPrepNow · Answer

n can equal 4, 20 or 100

Sum = 4 + 20 + 100 = 124

Answer: E

GMATWhizTeam · Answer

Solution:

We are given  LCM (n, 50) = 100

Or,  LCM (n, 2	imes 5^2)=2^2	imes 5^2

So for this to be true,  n  has to have  2^2  in its prime factorization because 50 doesn't have it.

Thus the possible values of  n = 2^2=4, 2^2	imes 5^1=20  and  2^2	imes 5^2=100 . Cannot go beyond this because then the LCM will change.

So, the sum of all possible values  = 4+20+100=124

Hence the right answer is    Option E   .

Bunuel · Answer

This question is a part of  Are You Up For the Challenge: 700 Level Questions  collection.

BrushMyQuant · Answer

Given that LCM of n and 50 is 100 and we need to find what is the sum of all possible values n

If we do prime factorization of 100 then we get  2^2 * 5^2

Now, prime factorization of 50 is  2 * 5^2 , this means that n should be either of the following

2^2  or  2^2 * 5  or  2^2 * 5^2  to make LCM(n,50) = 100

=>  n can be 4, 20 or 100

=> Sum of all possible values of n = 4 + 20 + 100 = 124

So ,  Answer will be E 
Hope it helps!

To learn more about LCM and GCD watch the following videos

https://www.youtube.com/watch?v=7jPV6JD0908

https://www.youtube.com/watch?v=vtSsAjcF-0Y

RenB · Answer

Can anyone share a link, if any, of questions of this type?

Bunuel · Answer

Somewhat similar questions: https://gmatclub.com/forum/if-the-least ... 65381.html https://gmatclub.com/forum/the-least-co ... 63582.html https://gmatclub.com/forum/if-the-least ... 78148.html

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If the least common multiple of a positive integer n and 50 is 100, wh

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