OlympiadHardNumber Theory10–12

For how many integers $n$ with $1 \le n \le 100$ is $n^{2} + n + 1$ divisible by $3$ ?

Para quantos inteiros $n$ com $1 \le n \le 100$ é que $n^{2} + n + 1$ é divisível por $3$ ?

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Solution

Test the three residues: $n \equiv 0$ gives $1$ , $n \equiv 1$ gives $3 \equiv 0$ , $n \equiv 2$ gives $7 \equiv 1$ . Only $n \equiv 1 \pmod{3}$ works, that is $1, 4, \ldots, 100$ — $34$ values.

For how many integers nnn with 1≤n≤1001 \le n \le 1001≤n≤100 is n2+n+1n^{2} + n + 1n2+n+1 divisible by 333?

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For how many integers $n$ with $1 \le n \le 100$ is $n^{2} + n + 1$ divisible by $3$ ?