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

Find the maximum value of n to the power 1/n and prove that it is a maximum.

What are the possible remainders when the 100-th power of an integer is divided by 125?

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)