Question

How many pairs (m, n) of positive integers satisfy the equation m2 + 105 = n2? (type in box)

Answer 1

Calculation:

m² + 105 = n²

⇒ n² – m² = 105

⇒ (n – m)(n + m) = 105

n, m are positive integer

Factor of 105 = 1, 3, 5, 7, 15, 21, 35, 105

⇒ n + m > n – m

Condition,

n + m = 105 & n – m = 1

Solve m & n

⇒ n = 53, m = 52

n + m = 35 & n – m = 3

Solve m & n

⇒ n = 19, m = 16

n + m = 21 & n – m = 5

Solve m & n

⇒ n = 13, m = 8

n + m = 15 & n – m = 7

Solve m & n

⇒ n = 11, m = 4

∴ There are four pair.