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 $3. what are all possible integers dorothy may have input? i got the answer

The correct answer is:

12.

Explanation:

Let x be the integer she inputs.

"One less than triple the input" would be 3x-1.

She gets 3 as the output; this means 3x-1=3.

To solve this, add 1 to each side:
3x-1+1=3+1
3x=4

Divide both sides by 3:
3x/3 = 4/3
x=4/3 = 1 1/3.

This will not work, however, as it is not an integer.

"One more than half the number" is 1/2x+1.  Since she gets 3 as the output, this gives us:
1/2x+1=3

Subtract 1 from each side:
1/2x+1-1=3-1
1/2x=2

Divide both sides by 1/2:


When dividing fractions, flip the second fraction and multiply:
x=2÷(1/2) = 2×(2/1) = (2/1)×(2/1) = 4/1 = 4

While this is an integer, this does not work either, because this expression is for inputs that are even but not divisible by 4.  This is divisible by 4, so it is not accurate.

"One-fourth of the input" is 1/4x.  The output is 3, which gives us:
1/4x =3

Dividing both sides by 1/4, we have:


When dividing fractions, we flip the second fraction and multiply:
x=3÷(1/4) = 3×(4/1) = (3/1)×(4/1) = 12/1 = 12

This expression is for an input that is divisible by 4.  12 is divisible by 4, so this is accurate.

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