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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.

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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

### Key Concepts

Integers

Complex Numbers

Sets

But try the problem first…

Source

AIME I, 1990, Question 10

Complex Numbers from A to Z by Titu Andreescue

## Try with Hints

First hint

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

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

Second Hint

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

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

Final Step

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

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

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