In the equation 1/j + 1/k = 1/8, both j and k are positive integers. What is the sum of all possible values for k?
+128474 j, k > 8
Let 8 =z
Let j = z + a
Let k = z + b
So we have
1/ ( z + a) + 1 / ( z + b) = 1/z
(2z + a + b) / ( z^2 + az + bz + ab) = 1 /z cross-multiply
z ( 2z + a + b) = z^2 + az + bz + ab
2z^2 + az + bz = z^2 + bz + az + ab
z^2 = ab
So
8^2 = ab
64 = ab
Factors of 64 = 1, 2 , 4 , 8 , 16 , 32 , 64
So
a b = (z + a) (z + b) = j , k
64 1 = 72 9 72 9
32 2 = 40 10 40 , 10
16 4 = 24 12 24 , 12
8 8 = 16 16 16 , 16
Depending on our choice for k the sum of all possible values =
(72 + 9 + 40 + 10 + 24 + 12 + 16) = 183
