The least common multiple of positive integers p and q, where p q

Question

The least common multiple of positive integers p and q, where p > q, is 935. What is the maximum possible sum of the digits of q?

A. 1
B. 2
C. 8
D. 13
E. 15

ArunSharma12 · Answer

935 = 5*11*17
since P > Q, Q can not be 935.
possible values of Q  
maximum possible sum of the digits = 16.
but 16 is not among the options? is there a different logic to answer this question?

rajatchopra1994 · Answer

Dear  ArunSharma12  ;

Please find the explanation below:

We know, 935 = 5*11*17

Let p = hx
q = hy
Where h is the hcf of p and q.
Therefore, lcm of p and q= hxy

We have lcm = 935.
hxy = 935

If h = 1,
Then p = 935
q = 1
Sum of digits of q = 1

If h = 5,
p = 5*17 = 85
q = 5*11 = 55
Sum of digits of q = 10

If h = 11,
p = 11*17 = 187
q = 11*5 = 55
Sum of digits of q = 10

If h = 17,
p = 17*11 = 187
q = 17*5 = 85
Sum of digits of q = 13

Maximum possible sum of digits of q = 13.

IMO-D

ArunSharma12 · Answer

taking the logic mentioned in your post further 
h = 187
P =5*h = 5*11*17 = 935
Q= 1*h = 11*17 = 187
LCM will still be 935 (5*11*17)
can you explain why this case is not valid for this question statement?

fireagablast · Answer

the problem states the p>q, so q should be the smaller number here. 
But even though q is the smaller number, we are looking for the LCM such that q is at it's max.

You want to pick the two largest value such that their prime factors are most distinct.
187 = 17*11
85 = 17*5
LCM(187,85) = 17*11*5 = 935

p=187
q=85
8+5=13

chetan2u · Answer

Given that LCM(p,q)=935 and p>q

If  p=935=5*11*17 , q could be any combinations of factors of 935, because the LCM of a number and any of its factors will always be that number itself..

So, the possible values of q are
1)  5 ...5,
2)  11 ....1+1=2,
3)  17 .....1+7=8, 
4)  5*11=55 ....5+5=10, 
5)  5*17=85 .....8+5=13, 
6)  11*17=187 ......1+8+7=16
We cannot have q as 935 as p>q

So answer is 16

However  if it was given 935>p>q , p would be 187, and q would be 85. Thus, the answer would be 13 in that case.

But here the answer is 16.

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The least common multiple of positive integers p and q, where p > q

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