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Pythagorean Golden Means

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

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There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

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Mod 7

Find the remainder when 3^{2001} is divided by 7.

The Public Key

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
This problem is intended to be tackled in conjunction with reading the article on Public Key Cryptography though it is not essential to understand the details of Public Key Cryptography in order to do the problem. In itself it is a challenge to systematically reduce the large number $180^{59}$ (using modulus arithmetic) to one which the calculator can handle and finally to its equivalent modulo 391.