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Least possible value of r+s

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Least possible value of r+s

by gmattesttaker2 » Sat Feb 22, 2014 12:08 am
Hello,

Can you please assist with this:

lf r and s are positive integers, each greater than 1 and if 11(s - l) = l3(r - l), what is the least possible value of r + s?

A) 2
B) 11
C) 22
D) 24
E) 26

OA: E


I tried to solve this as follows:

11(s - l) = l3(r - l)

=> 11s - 11 = 13r - 13
=> -11 + 13 = 13r - 11s
=> 2 = 13r - 11s

Given r,s > 1

I then started testing the answer choices as follows:

Let, r + s = 22 => r = 22 - s

So, 11 ( s - 1 ) = 13 ( 22 - s - 1)
=> 11s - 11 = 13 ( 21 - s )
=> 11s - 11 = 273 - 13s
=> 11s + 13s = 273 + 11
=> 24s = 284
=> s = 284/24 = not an integer

Hence, r + s cannot be 22

I was wondering if there is a better way to solve this problem?

Thanks a lot,
Sri
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by Uva@90 » Sat Feb 22, 2014 1:47 am
gmattesttaker2 wrote:Hello,

Can you please assist with this:

lf r and s are positive integers, each greater than 1 and if 11(s - l) = l3(r - l), what is the least possible value of r + s?

A) 2
B) 11
C) 22
D) 24
E) 26

OA: E


I tried to solve this as follows:

11(s - l) = l3(r - l)

=> 11s - 11 = 13r - 13
=> -11 + 13 = 13r - 11s
=> 2 = 13r - 11s

Given r,s > 1

I then started testing the answer choices as follows:

Let, r + s = 22 => r = 22 - s

So, 11 ( s - 1 ) = 13 ( 22 - s - 1)
=> 11s - 11 = 13 ( 21 - s )
=> 11s - 11 = 273 - 13s
=> 11s + 13s = 273 + 11
=> 24s = 284
=> s = 284/24 = not an integer

Hence, r + s cannot be 22

I was wondering if there is a better way to solve this problem?

Thanks a lot,
Sri
Sri,
You ended up here,
13r - 11s = 2 => r+s +12r-12s =2 => r+s +12(r-s) =2
=> r+s = 2+12(s-r) => 2(1+6(s-r))

r+s = 2(1+6 (s-r) )

Let S-r = 0
r+s =2. But Question States that both r ans are greater than 1 hence their sum must be >2 REJECTED

Let S-r =1
r+s = 2(1+ 6(1)) =14 But Not in option. REJECTED.

Let S-r =2
r+s = 2(1+ 6(2)) = 26.YES it is there
Hence ans is E

Regards,
Uva.
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by GMATGuruNY » Sat Feb 22, 2014 3:56 am
gmattesttaker2 wrote:Hello,

Can you please assist with this:

lf r and s are positive integers, each greater than 1 and if 11(s - l) = l3(r - l), what is the least possible value of r + s?

A) 2
B) 11
C) 22
D) 24
E) 26

OA: E
11(s - 1) = 13(r - 1) = x.
To minimize the values of r and s, we must minimize the value of x.
Since r and s are integers, x must be a multiple of the factors in red.
Thus, the least possible value for x = 11*13.
If x = 11*13, we get:

11(s-1) = 11*13
s-1 = 13
s=14.

13(r-1) = 11*13
r-1 = 11
r=12.

Thus, the least possible value for r+s = 12+14 = 26.

The correct answer is E.
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