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

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

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