Problem Solving - Algebra

[xtremecoder007]

If x and Y are positive integers and 1 + x + y + xy = 15. what is x + y?

[Anurag@Gurome]

If x and Y are positive integers and 1 + x + y + xy = 15. what is x + y?

(1 + x + y + xy) = (1 + x)(1 + y)

As x and y are positive integers, (x + 1) and (y + 1) will be also positive integers.

Now 15 can be factorized as any of the following ways,

• 1. 15 = 1*15 = 15*1
  2. 15 = 3*5 = 5*3

If we take 15 = 1*15 or 15*1, then either of (x + 1) or (y + 1) must be equal to 1. Hence, either of x or y must be equal to 0 which is not a positive integer. Hence, this pair is not a valid pair.

But if we take 15 = 3*5 or 5*3, then either (x + 1) or (y + 1) will be equal to 3 and the other one will be equal to 5. Hence, either x or y will be equal to 2 and the other one will be equal to 4. Therefore, (x + y) will always be equal to (2 + 4) = 6

Hence, (x + y) = 6

[GMATGuruNY]

If x and Y are positive integers and 1 + x + y + xy = 15. what is x + y?

x + y + xy = 14.

If x=1, we get:
1 + y + y = 14.
y = 13/2.
Doesn't work.

If x=2, we get:
2 + y + 2y = 14.
y = 4.
Success!

x+y = 2+4 = 6.