A number is divisible by \(3\) if the sum of its digits is divisible by \(3.\)

If \(n \ge 1,\) then \(6 \mid n^3 + 3n^2 + 2n.\)

If \(n \ge 1,\) then \(a + b \mid a^{2n + 1} + b^{2n + 1}.\)

If \(n \ge 2,\) then \(x - 1 \mid x^n - 1.\)

If \(n \ge 1,\) then \(3 \mid 2^{2n} - 1.\)

If \(n \ge 0,\) then \(7 \mid 3^{2n + 1} + 2^{n + 2}.\)

If \(n \ge 1,\) then \(3 \mid n^3 - n.\)

If \(n \ge 1,\) then \(3 \mid n^3 + 2n.\)

If \(n \ge 1,\) then \(x - 1 \mid x^n - 1.\)

If \(n \ge 1,\) then \(a - b \mid a^n - b^n.\)

If \(n \ge 1,\) then \(2 \mid n^2 - n + 2.\)

If \(n \ge 1,\) then \(2 \mid n^2 - n + 2.\)

If \(n \ge 1\), then \(2 \mid n^2 + n.\)

If \(n \ge 1\), then \(2 \mid 3^n - 1.\)

If \(n \ge 1\), then \(3 \mid 5^n - 2^n.\)

If \(n \ge 1\), then \(7 \mid 8^n - 1.\)

If \(n \ge 1\), then \(6 \mid 7^n + 5.\)

If \(n \ge 1\), then \(4 \mid 6^n - 2^n.\)

If \(n \ge 1\), then \(3 \mid 4^n - 1.\)

If \(n \ge 1\), then \(6 \mid n(n^2 + 5).\)

If \(n \ge 1\), then \(17 \mid 18^n - 1.\)

If \(n \ge 1\), then \(7 \mid 11^n - 4^n.\)

If \(n \ge 1\), then \(3 \mid n^3 + 2n.\)

If \(n \ge 1\), then \(8 \mid 3^{2n} - 1.\)

If \(n \ge 1\), then \(80 \mid 3^{4n} - 1\).

Let \(n\) be a positive odd number. Prove that \(10 \mid 3^n + 7^n\).

The sum of the cubes of three consecutive positive integers is divisible by 9.