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

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Bunuel
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The least common multiple of positive integers p and q, where p > q [#permalink] New post   13 Apr 2020, 10:31
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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

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ArunSharma12
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Re: The least common multiple of positive integers p and q, where p > q [#permalink] New post   13 Apr 2020, 11:13
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Bunuel wrote:
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


935 = 5*11*17
since P > Q, Q can not be 935.
possible values of Q [5,11,17,55,85,187]
maximum possible sum of the digits = 16.
but 16 is not among the options? is there a different logic to answer this question?
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rajatchopra1994
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Re: The least common multiple of positive integers p and q, where p > q [#permalink] New post   13 Apr 2020, 11:41
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ArunSharma12 wrote:
Bunuel wrote:
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


935 = 5*11*17
since P > Q, Q can not be 935.
possible values of Q [5,11,17,55,85,187]
maximum possible sum of the digits = 16.
but 16 is not among the options? is there a different logic to answer this question?


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
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Re: The least common multiple of positive integers p and q, where p > q [#permalink] New post   13 Apr 2020, 20:55
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rajatchopra1994 wrote:

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


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?
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Re: The least common multiple of positive integers p and q, where p > q [#permalink] New post   13 Apr 2020, 20:57
ArunSharma12 wrote:
Bunuel wrote:
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


935 = 5*11*17
since P > Q, Q can not be 935.
possible values of Q [5,11,17,55,85,187]
maximum possible sum of the digits = 16.
but 16 is not among the options? is there a different logic to answer this question?


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
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Re: The least common multiple of positive integers p and q, where p > q [#permalink] New post   13 Apr 2020, 21:34
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Bunuel wrote:
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



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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Re: The least common multiple of positive integers p and q, where p > q [#permalink] New post   17 May 2021, 18:46
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Re: The least common multiple of positive integers p and q, where p > q [#permalink]
17 May 2021, 18:46
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