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

CalculatorUser Jan 30, 2019

edited by
Guest
Jan 30, 2019

edited by Guest Jan 30, 2019

edited by Guest Jan 30, 2019

#2**+1 **

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

CPhill Jan 30, 2019