Mock AIME I 2015 Problems/Problem 1
David, Justin, Richard, and Palmer are demonstrating a "math magic" concept in front of an audience. There are four boxes, labeled A, B, C, and D, and each one contains a different number. First, David pulls out the numbers in boxes A and B and reports that their product is . Justin then claims that the product of the numbers in boxes B and C is , and Richard states the product of the numbers in boxes C and D to be . Finally, Palmer announces the product of the numbers in boxes D and A. If is the number that Palmer says, what is ?
Associative Property! Note that the product of all four numbers can be written in two different ways, and . Setting these equal to each other gives