Let f(x) be a function defined on set of non negative integers and taking values in the same set such that f satisfies:

x – f(x) = 19[x/19] – 90[f(x)/90] and, 1900< f(1990) < 2000.

Determine the possible value(s) that f(1990) can take.

NOTE: [n] denotes the greatest integer < = n

Rearrange the terms to

90[f(x)/90] - f(x) = 19[x/19] - x

and substitute

90[f(1990)] - f(1990) = 1976 - 1990 = -14

What this says is that f(1990) is 14 more than a multiple of 90, or f(1990) = 90 k + 14, where k is an integer.

The lower limit of possible answers, 1900, is 10 more than a multiple of 90, so 1904 is 14 more than a multiple of 90, and so is the first possibility. The next possibility would be 1904+90 = 1994.

Thus the two possibilities are 1904 and 1994.

Posted by Charlie on 2007-04-28 10:10:39