Problem Solving

sureshtx wrote:If x and y are positive integers and 1 + x + y + xy = 15, what is the value of x + y?
A. 3
B. 5
C. 6
D. 8
E. 9

We have

x+y+xy = 14

That's a very small sum of three positive integers, so there wouldn't be much work involved in trying numbers until we found the answer. Still, we can save some time by noticing that x and y both must be even; if x and y were both odd, the sum would be odd, and if one were even, the other odd, that would make xy even, and the sum would again be odd. Once we see that x and y are both even, and that x=2, y=2 gives too small a sum, you find that one of them is 2, the other 4, and x+y = 6.

Alternatively, you could notice:

1 + x + y + xy = 15
1 + x + (1+x)y = 15
(1+x)(1+y) = 15

and since 15 only has two proper divisors, we know that 1+x is equal to either 3 or 5, and 1+y is either 5 or 3. So x and y are 2 and 4 (or 4 and 2), and x+y = 6.