jmc

Question

A binary operation has the properties that a,, (b, ,c) = (a, ,b) c and that a, ,a=1 for all nonzero real numbers a, b, and c. (Here represents multiplication). Find the solution to the equation 2016 ,, (6,, x)=100.

Accepted Answer

Setting b = a and c = a, we get a , , (a , , a) = (a , , a) a,which reduces to a , , 1 = a for any nonzero a. Setting c = b, we get a , , (b , , b) = (a , , b) b,which reduces to a , , 1 = (a , , b) b, so a = (a , , b) b. Hence, a , , b = abfor any nonzero a and b. We want to solve 2016 , , (6 , , x) = 100, or 20166x = 100.Solving, we find x = 2584.