1/a + 1/b = 1/18

(a + b)/(ab) = 1/18

a + b = 18/(ab)

Note that:

(a - n)(b - n) = ab - n(a + b) + n²

Substitute 18/(ab) for a + b:

(a - n)(b - n) = ab - 18n/(ab) + n²

The RHS simplifies:

(a - n)(b - n) = ab(18 - n)/18 + n²

If we choose n = 18, the first term on the RHS goes to zero.

(a - 18)(b - 18) = 324

The factors of 324 are:

(1)(324)
(2)(162)
(3)(108)
(4)(81)
(6)(54)
(9)(36)
(12)(27)
(18)(18)

Adding 18 to each of these factors gives us eight pairs of integers for a and b that satisfy the original equation.

1/19 + 1/342 = 1/18

1/20 + 1/180 = 1/18

1/21 + 1/126 = 1/18

1/22 + 1/99 = 1/18

1/24 + 1/72 = 1/18

1/27 + 1/54 = 1/18

1/30 + 1/45 = 1/18

1/36 + 1/36 = 1/18

View other answers by : Soroban