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Note the function, get the value(s) (Posted on 2007-04-28) Difficulty: 3 of 5
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

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution solution | Comment 1 of 2

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
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