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# Complex Numbers and prime | AIME I, 2012 | Question 6

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Complex Numbers and prime.

## Complex Numbers and primes - AIME 2012

The complex numbers z and w satisfy $z^{13} = w$ $w^{11} = z$ and the imaginary part of z is $\sin{\frac{m\pi}{n}}$, for relatively prime positive integers m and n with m<n. Find n.

• is 107
• is 71
• is 840
• cannot be determined from the given information

### Key Concepts

Complex Numbers

Algebra

Number Theory

AIME I, 2012, Question 6

Complex Numbers from A to Z by Titu Andreescue

## Try with Hints

First hint

Taking both given equations $(z^{13})^{11} = z$ gives $z^{143} = z$ Then $z^{142} = 1$

Second Hint

Then by De Moivre's theorem, imaginary part of z will be of the form $\sin{\frac{2k\pi}{142}} = \sin{\frac{k\pi}{71}}$ where $k \in {1, 2, upto 70}$

Final Step

71 is prime and n = 71.