Question

Prove that among any three consecutive integers, one of them is a multiple of 3.

Answer 1

Hint: Let the three consecutive integers be n, n + 1, and n + 2. What are the possible
values of n mod 3? What does this translate into, according to the division algorithm? In
each case, what would n, n + 1, and n + 2 look like?

Answer 2

suppose the smallest is of the form n = 3k
Then clearly it is divisible by 3 -- that is it is congruent to 0 (mod 3)
Then the next two numbers are congruent to 1 and 2.

In fact, any number when divided by 3 will leave a remainder of 0, 1, or 2.

So no matter which residue is left by the smallest, the next two numbers will exhaust the other two possibilities/ That is, one of them will be leave a remainder of 0 -- that is, it is a multiple of 3.