The product of the squares of two consecutive integers is 256036. Find the integers.

Question

I need a walk through on the problem, please.

Answer 1

Suppose that k is a positive integer. By assumption we have k^2(k+1)^2=256036=(506)^2 from where we obtain that the positive integer k needs to satisfy the equation k(k+1)=506 which is the same as k^2+k-506=0. This quadratic equation has two solutions k_1=-23 and k_2=22. Since we assumed the integer k to be positive we obtain the numbers 22 and 23 that are consecutive and satisfy (22^2)(23)^2=256036.