Solution to Problem #81


The woman's third deposit is x + y, her fourth 2x + y, her fifth 3x + 2y (in general her nth is Fn-2 x + Fn-1 y), so her 20th deposit is 2584x + 4181y = 1000000. We must find positive integer solutions to this equation.

One of the basic properties of Fibonacci numbers is that Fn2 - Fn-1Fn+1 = (-1)n+1. Taking n = 19, we get 2584(-2584) + 4181(1597) = 1 and multiplying both sides by 1000000, we have 2584(-2584000000) + 4181(1597000000) = 1000000, so x0 = -2584000000, y0 = 1597000000 gives an integer solution to our equation, but not a positive one. To obtain a positive solution we note that x = x0 + 4181t, y = y0 - 2584t will also be a solution to our equation for any integer t. Taking t = 618034 yields x = 154, y = 144.


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