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Question

If a, b, and c are integers satisfying a + 1b = 227, b + 1c = 8, and abc = 21, then find c + 1a. Express your answer as a common fraction.

Accepted Answer

Let x = c + 1a. Multiplying to take advantage of the symmetry, align* 227 8 x &= (a + 1b)(b + 1c)(c + 1a) &= abc + a + b + c + 1a + 1b + 1c + 1abc &= 21 + (a + 1b) + (b + 1c ) + (c + 1a) + 121 &= 21 + 227 + 8 + x + 121 &= 29 21 + 22 3 + 121 + x align* Thus, 22 8 321 x = 29 21 + 22 3 + 121 + x x = 29 21 + 22 3 + 122 8 3 - 21 = 676507 = 43.