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The article 'Public Key Cryptography ' gives a detailed explanation of how the method works and gives you help in working with modulus arithmetic.
This problem simply asks you to work out 180^{59}\pmod {391} which has to be done in stages because calculators and computers will not handle big numbers like 180^{59}.
David of Colyton Grammar School gave the very neat solution you see below and also wrote a program to check the answer and to calculate quickly other high powers in modulus arithmetic. Try out David's program.
In order to calculate 180^{59}\pmod {391} you have to find a sensible way to break down 180^{59} into pieces which can all be tackled individually. The easiest way of doing this I think is to first write 59 as a sum of powers of 2: 59 = 32+16+8+2+1
Now you know that 180^{59} = 180 \times 180^2 \times 180^8 \times 180^{16} \times 180^{32}. This is very useful because in the sequence 180, 180^2, 180^4, 180^8, 180^{16}, 180^{32} each term is the square of the previous term.
You also need to appreciate that in modular arithmetic: ab\pmod{c} = [a\pmod {c}]\times [b\pmod {c}]. This makes the numbers that you are going to have to deal with far more manageable and the problem can be written like this: 180 = 180 \pmod {391}
I have also written a console application to check my answer and to quickly calculate any other similar problems. It can do any number and mod with powers up to 255. As it was written in a rush it may still be slightly buggy but seems to work well every time I've tried it ;)
Andrei from Bucharest Romania gave a method which depends on expressing 180^{59} as a product and working out the factors separately. 180^{59}=180^{56}\times 180^3 = (180^{14})^4\times 180^3.
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.
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?