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# Complex numbers and Sets | AIME I, 1990 | Question 10

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Complex numbers and Sets.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Complex Numbers and Sets.

## Complex Numbers and Sets – AIME I, 1990

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

• is 107
• is 144
• is 840
• cannot be determined from the given information

Integers

Complex Numbers

Sets

## Check the Answer

AIME I, 1990, Question 10

Complex Numbers from A to Z by Titu Andreescue

## Try with Hints

18th and 48th roots of 1 found by de Moivre’s Theorem

=$cis(\frac{2k_1\pi}{18})$ and $cis(\frac{2k_2\pi}{48})$

where $k_1$, $K_2$ are integers from 0 to 17 and 0 to 47 and $cis \theta = cos \theta +i sin \theta$

zw= $cis(\frac{k_1\pi}{9}+\frac{k_2\pi}{24})=cis(\frac{8k_1\pi+3k_2\pi}{72})$

and since the trigonometric functions are periodic every period ${2\pi}$

or, at (72)(2)=144 distinct elements in C.