+2862
The sum of a positive integer and 100 is a square number. The sum of the same positive integer and 168 is a square number. Find the positive integer.
This is a chinese middle school problem
+128475 Here is the solution
N + 100 = P^2
N + 168 = Q^2
Subtract the first from the second and we get that
Q^2 - P^2 = 68
(Q + P) (Q - P) = 68
The factors that multiply to 68 are
68 1
34 2
17 4
But note that we cannot add/subtract two positive integers and obtain different parities
For example 5 + 2 = 7 and 5 - 2 = 3 ......both are odd
Then....the only possible solution is Q + P = 34 and Q - P = 2
Adding these we have that 2Q = 36 ⇒ Q = 18
And P = 16
So Q^2 = 324 and P^2 = 256
So.....using N + 100 = P^2 ⇒ N + 100 = 256 ⇒ N = 156
And checking
156 + 168 = 324 = Q^2 = 18^2
