2021 Fall AMC 12B Problems/Problem 16
Problem
Suppose ,
,
are positive integers such that
and
What is the sum of all possible distinct values of
?
Solution
Let ,
,
. WLOG, let $x\ley\lez$ (Error compiling LaTeX. Unknown error_msg). We can split this off into cases:
x=1, y=1, z=7: let a = 7A, c=7C, we can try all possibilities of A and C to find that a=7, b=9, c=7 is a solution.
x=1, y=2, z=6: No solutions. By y and z, we know that a, b, and c have to all be even. Therefore, x cannot be equal to 1.
x=1, y=3, z=5: C has to be both a multiple of 3 and 5. Therefore, c has to be a multiple of 15. The only solution for this is a=5, b=3, c=15.
x=1, y=4, z=4: No solutions. By y and z, we know that a, b, and c have to all be even. Therefore, x cannot be equal to 1.
x=2, y=2, z=5: No solutions. By x and y, we know that a, b, and c have to all be even. Therefore, z cannot be equal to 1.
x=2, y=3, z=4: No solutions. By x and z, we know that a, b, and c have to all be even. Therefore, y cannot be equal to 1.
x=3, y=3, z=3: No solutions. As a, b, and c have to all be divisible by 3, a+b+c has to be divisible by 3. This contradicts the sum .
Putting these solutions together, we have