Create the equations
One solution of the equation is .
If , , , and are all integers—not necessarily distinct—find the possible values of .
Give the sum of all of these values.
The answer is 252.
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1 solution
The four factors of 25 must be one of ( P , Q , R , S ) = ( 5 , 5 , − 1 , − 1 ) , ( 5 , 5 , 1 , 1 ) , ( − 5 , − 5 , 1 , 1 ) , ( − 5 , − 5 , − 1 , − 1 ) , ( 5 , − 5 , 1 , − 1 ) , ( 2 5 , 1 , 1 , 1 ) , ( 2 5 , − 1 , − 1 , 1 ) , ( − 2 5 , − 1 , 1 , 1 ) , ( − 2 5 , − 1 , − 1 , − 1 )
P + Q + R + S could be any of ( 8 , 1 2 , − 8 , − 1 2 , 0 , 2 8 , 2 4 , − 2 4 , − 2 8 )
However,
P + Q + R + S = ( 7 − a ) + ( 7 − b ) + ( 7 − c ) + ( 7 − d )
which can be rewritten as
a + b + c + d = 2 8 − ( P + R + S + T )
so
a + b + c + d could be any of ( 2 0 , 1 6 , 3 6 , 4 0 , 2 8 , 0 , 4 , 5 2 , 5 6 )
These sum to 2 5 2 .