Question

HOW CAN U PROVE THAT 6^100 ENDS WITH 6? ......

Answer 1

We can prove it using mathematical induction.

Claim: 6n≡6mod106n≡6mod10 for all n∈Nn∈N (the symbol NN denotes the natural numbers, and mod10mod10means we are using modular arithmetic with a modulus of 10).

Base case (i.e., showing it's true for n=1n=1):

61≡6mod10✓61≡6mod10✓

Induction step (i.e., showing that, if it is true for n=kn=k, then it is true for n=k+1n=k+1):

6k≡6mod10⟹6k+1≡6k⋅6≡6⋅6≡36≡6mod10