Problem #2

Question: say whether the following is true or false and support your answer by a proof: The sum of any five consecutive integers is divisible by 5 (without reminder).
Answer: it's true.

Claim: the sum of any five consecutive integers is divisible by 5.
Proof: let's prove it directly. Let's take an arbitrary number $n \in {Z}$. Then the sum of five consecutive integers can be presented as:

$$n+(n+1)+(n+2)+(n+3)+(n+4) = 5n+10 = 5(n+2)$$

By the definition of divisibility $5(n+2)$ is divisible by 5, so the sum of any five consecutive integers is divisible by 5. $\blacksquare$