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If you take any integer nn and calculate n3βˆ’nn^3 - n, the result is always divisible by 66. Try it with n=5n = 5: 125βˆ’5=120125 - 5 = 120, which is 6Γ—206 \times 20.
Testing numbers one by one cannot prove this works for all infinite integers. To build unbreakable proofs, we rely on the formal mechanics of divisibility and modular arithmetic.
How do mathematicians define divisibility without relying on decimals or fractions?

The Definition of Divisibility

For any integers aa and bb where aβ‰ 0a \neq 0, we say aa divides bb (written a∣ba \mid b) if there exists an integer kk such that b=aβ‹…kb = a \cdot k.
The vertical bar ∣\mid represents a relation that is either true or false, not an operation like division (b/ab / a).
A fundamental theorem is linear combination divisibility: if a∣ba \mid b and a∣ca \mid c, then a∣(bx+cy)a \mid (bx + cy) for any integers xx and yy.
Why does this hold? Since b=ak1b = a k_1 and c=ak2c = a k_2, substituting gives bx+cy=a(k1x+k2y)bx + cy = a(k_1 x + k_2 y), and the expression in parentheses is guaranteed to be an integer.
What happens when we only care about the remainder left behind after division?

Modular Congruence

In 1801, nineteen-year-old Carl Friedrich Gauss introduced modular congruence in his landmark work Disquisitiones Arithmeticae to turn questions about remainders into standard algebra.
We say aa is congruent to bb modulo mm (written a≑b(modm)a \equiv b \pmod m) if m∣(aβˆ’b)m \mid (a - b), which means aa and bb leave the same remainder when divided by the positive integer mm.