Given positive integers x and y such that x is not equal to y and 1/x + 1/y = 1/24, what is the smallest possible positive value for x + y?
Let z = 24
Then x must be z + a where a > 0
And y must be z + b where b > 0 ....so we have
1/ (z + a) + 1 / (z + b) = 1/ z
( z + a + z + b) / [ (z + a) (z + b) ] = 1/ z
( a + b + 2z) / ( z^2 + az + bz + ab) = 1 / z cross-multiply
az + bz + 2z^2 = z^2 + az + bz + ab
z^2 = ab
24^2 = ab
576 = ab
Factors of 576 =
1 | 2 | 3 | 4 | 6 | 8 | 9 | 12 | 16 | 18 | 24 | 32 | 36 | 48 | 64 | 72 | 96 | 144 | 192 | 288 | 576
18 and 32 are the two factors that will lead to minimum values for x and y
Let a = 18 and b = 32
x = z + a = 24 + 18 = 42
y = z + b = 24 + 32 = 56
x + y = 98