Introduction to Mathematical Thinking-Problem 5

5.Prove that for any integer $n$n, at least one of the integers $n, n+2, n+4$n,n+2,n+4 is divisible by 3.

Proof: By the Division Theorem, $n$n can be expressed as either $3q$3q, or $3q+1$3q+1, or $3q+2$3q+2, where $q$q is an integer.
If $n=3q$n=3q, we have $n$n divisible by 3.
If $n=3q+2$n=3q+2, we have $n+4=3q+6$n+4=3q+6 divisible by 3.
If $n=3q+4$n=3q+4, we have $n+2=3q+6$n+2=3q+6 divisible by 3.
Therefore, for any integer $n$n, at least one of the integers $n, n+2, n+4$n,n+2,n+4 is divisible by 3.

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