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What is the number of pairs of non-zero integers whose sum is 72 more than their product is?
Wiki User
∙ 14y ago
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.
Wiki User
∙ 14y ago
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