### Instruction: 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?### Response: To find Ramanujan's number, I need to divide the product by Hardy's number.
I can do this using complex fraction arithmetic, but I need to multiply the numerator and denominator by the conjugate of the denominator to make it real.
The conjugate of $5+3i$ is $5-3i$, so I multiply both the numerator and denominator by that.
The numerator becomes $(32-8i)(5-3i) = 160-96i-40i+24i^2 = 136-136i$.
The denominator becomes $(5+3i)(5-3i) = 25-15i+15i-9i^2 = 34$.
So the quotient is $\frac{136-136i}{34} = 4-4i$.
This means Ramanujan's number was $4-4i$.