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?
+248 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.
+248 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.
