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Question

Ramanujan and Hardy played a game where they both picked a complex number. If the product of their numbers was 32-8i, and Hardy picked 5+3i, what number did Ramanujan pick?

Accepted Answer

Let Hardy's number be h and Ramanujan's be r. We have the equations: align* rh&=32-8i, h&=5+3i. align* Thus, r=32-8i5+3i. Multiplying top and bottom by the conjugate of 5+3i, we have r=(32-8i)(5-3i)34 or r=136-136i34=4-4i