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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.

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

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

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 $k \in {1, 2, upto 70}$

71 is prime and n = 71.