Question

Dorothy has a mysterious $?$ button on her calculator. When she types in an integer and hits the $?$ button, if the input is odd, the calculator outputs $1$ less than triple the input. if the input is even but not divisible by $4$, the calculator outputs $1$ more than half the number. if the input is divisible by $4$, the calculator outputs one-fourth of the input. Dorothy typed in an integer, hit the $?$ button, and saw an output of $13$. What are all possible integers Dorothy may have input?

Answer 1

{52}

Explanation:

The calculator function appears to be ...

f(x) = {3x -1, x odd; x/2 +1, x not divisible by 4; x/4, x divisible by 4}

The inverse of the first function is ...

x = 3y -1

(x+1)/3 = y . . . . (y must be odd)

For x = 13, this is 14/3, which is not an integer.

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The inverse of the second function is ...

x = y/2+1

2(x-1) = y . . . . (y must not be divisible by 4)

For x = 13, this is 2·12 = 24, which is divisible by 4, so 24 is not the input value.

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The inverse of the third function is ...

x = y/4

4x = y . . . . (y must be, and is, divisible by 4)

For x = 13, this is 4·13 = 52.

The only possible input value for an output of 13 is 52.