Question

The sets $A=\{z:z^{18}=1\}$ and $B=\{w:w^{48}=1\}$ are both sets of complex roots of unity. The set $C=\{zw:z\in A,\ w\in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C?$

Answer

Answer: 144

Concepts:roots of unity least common multiple complex number

Difficulty rating: 2270

Small Hint:

Write the roots as exponentials whose arguments are multiples of $\frac{2\pi}{18}$ and $\frac{2\pi}{48}$

Big Hint:

The sums of those arguments generate all multiples of $\frac{2\pi}{\operatorname{lcm}(18,48)}$

Solution:

The arguments of products in $C$ are $2\pi\left(\frac a{18}+\frac b{48}\right)=\frac{2\pi(8a+3b)}{144}.$ Since $\gcd(8,3)=1,$ the residues $8a+3b$ generate every residue modulo $144.$ Thus $C$ is precisely the set of $144$ th roots of unity. Equivalently, its order is $\operatorname{lcm}(18,48)=144.$

1990 AIME Problem 10

Answer: 144

Difficulty rating: 2270

Small Hint:

Big Hint:

Solution:

Problem 10 in Other Years