What is the number of pairs of non-zero integers whose sum is 72 more than their product is?

Answer 1

Let the integers be m and n.

∴ By the condition given in the question,

m + n - mn = 72

∴ m + n - mn = 71 + 1

∴ m + n - mn - 1 = 71

∴ (m - 1)(1 - n) = 71

∴ (m - 1)(1 - n) = 1 × 71 = 71 × 1 = (-1) × (-71) = (-71) × (-1)

Case 1:

m - 1 = 1 and 1 - n = 71

∴ (m, n) = (2, -70)

Case 2:

m - 1 = 71 and 1 - n = 1

∴ (m , n) = (72, 0)

This case is inadmissible as m, n are non zero integers.

Case 3:

m - 1 = -1 and 1 - n = -71

∴ (m , n) = (0, 72)

This case is inadmissible as m, n are non zero integers.


Case 4:

m - 1 = -71 and 1 - n = -1

∴ (m, n) = (-70, 2)

But as the given expression is symmetric, therefore, (2, -70) and (-70, 2) cannot be considered two different pairs.

∴ We get only one solution.