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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?

Jun 5, 2019

#1
+1

First write out all the cases:

Odd: 3x-1

Even(not divisible by 4): .5x+1

Divisible by 4: .25x

Dorthy could have entered any type of number, so set 13 equal to what the button could do to the number she entered:

13=3x-1

14=3x

14 is not divisible by 3, so she did not enter an odd number.

13=.5x+1

12=.5x

x=24

24 is divisible by 4 so she could not have entered a number only divisible by 2.

13=.25x

x=52 so \(\boxed{52}\) is the only number she could have plugged into this function.

Jun 5, 2019

#1
+1

First write out all the cases:

Odd: 3x-1

Even(not divisible by 4): .5x+1

Divisible by 4: .25x

Dorthy could have entered any type of number, so set 13 equal to what the button could do to the number she entered:

13=3x-1

14=3x

14 is not divisible by 3, so she did not enter an odd number.

13=.5x+1

12=.5x

x=24

24 is divisible by 4 so she could not have entered a number only divisible by 2.

13=.25x

x=52 so \(\boxed{52}\) is the only number she could have plugged into this function.

power27 Jun 5, 2019