AIME Problem: Sets A = {z^16 = 1} and B = {w^48 = 1} are....

For other readers:

The sets A = {z : z<sup>18</sup> = 1} and B = {w : w<sup>48</sup> = 1} are both sets of complex roots of unity. The set C = {zw: z is in A and w is in B} is also a set of complex roots of unity. How many distinct elements are in C?
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To Trenters4325: Are you familiar with complex numbers, set notation, etc, or do you need links to lessons on some or all of the background material?

In your reply, please include all of the steps and reasoning you have attempted thus far. Thank you.

Eliz.
 
We are given that:
\(\displaystyle \begin{array}{l}
A = \{ z:z^{18} = 1\} = \left\{ {\cos \left( {\frac{{k\pi }}{9}} \right) + i\sin \left( {\frac{{k\pi }}{9}} \right):k = 0,1...,17} \right\} \\
B = \{ z:z^{48} = 1\} = \left\{ {\cos \left( {\frac{{j\pi }}{{24}}} \right) + i\sin \left( {\frac{{j\pi }}{{24}}} \right):j = 0,1...,47} \right\} \\
C = \left\{ {\cos \left( {\frac{{\left( {8k + 3j} \right)\pi }}{{72}}} \right) + i\sin \left( {\frac{{\left( {8k + 3j} \right)\pi }}{{72}}} \right):k = 0,1...,17\;\& \;j = 0,1...,47} \right\} \\
\left\{ {\left( {8k + 3j} \right):k = 0,1...,17\;\& \;j = 0,1...,47} \right\} \\
\end{array}\).
Now it comes down to counting the numbers of distinct elements in that last set.
 
Yes, I am familiar with complex numbers and set notation.

But what are "complex roots of unity"?

I am asking because I want to understand the question before I attempt a solution.
 
Trenters4325 said:
But what are "complex roots of unity"?
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For each positive integer, n, there are n different complex numbers the nth power of which equals 1.
These are called the n nth roots of unity.
 
Wow, that last step is harder than I thought.

Its 48*18=864 less the number of extraneous ordered pairs (of k and j).
So, 8*3=24 is important here, but I'm not sure where to go from there.
 
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