Identities

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

By proving these particular identities, prove the existence of general cases.