Question

How many solutions (m, n) exist for the equation n = √(100 - m²) where both m and

n are integers?
(A) 4
(C) 7
(D) 8
(B) 6
(E) 10

Answer 1

A) 4

Explanation:

First note that m cannot exceed 10 otherwise 100-m² will be negative and √(100-m²) will not be a real number much less an integer

Enumerate all values of m from 0 through 10, find the square, subtract from 100 and see if the result is a perfect square.- we take the square root and see if it is an integer value. The table below details everything. m and n are integers on in 4 cases m = 0, 6, 8 or 10.

Hence the answer is 4 solutions

m m² 100-m² √(100-m²)

0 0 100 10

1 1 99

2 4 96 9.797958971

3 9 91 9.539392014

4 16 84 9.16515139

5 25 75 8.660254038

6 36 64 8

7 49 51 7.141428429

8 64 36 6

9 81 19 4.358898944

10 100 0 0