Divisibility: (1+x)^n - 1 is divisible by x for all positive integers n (all terms after the first contain x).

Proving divisibility: $n^p$ - n is divisible by p (prime) -- this is Fermat's little theorem. Binomial approach for small cases: 3^n - 2^n - 1 divisibility can be shown by expanding 3^n = (2+1)^n.

Last digits: To find last two digits of $a^n$, compute $a^n$ mod 100. Use binomial expansion with a decomposed near a multiple of 10 or 100.

Example: Last two digits of 3^{100}: 3^{100} = (3^2)^{50} = 9^{50} = (10-1)^{50}. Expanding: C(50,0)*10^{50} - C(50,1)*10^{49} + ... + C(50,48)*10^2 - C(50,49)*10 + 1. Last two digits from: -C(50,49)*10 + 1 = -500 + 1 = -499, last two digits = 01.