Cube of 13

13 cubed equals 2197. Cubes show up in volume calculations — a cube with side 13 holds 2197 cubic units.

Cubes carry the fingerprint of their base. The final digit of 2197 is 7 — always the cube of the base's own final digit. Divided by 9, a cube only ever leaves 0, 1 or 8, and 2197 leaves 1. The volume picture explains the name: a 13 × 13 floor of unit cubes holds 169 of them, and 13 such layers stack to 2197 — the shape itself is the exponent at work. The gaps grow like a parabola: 2197 − 1728 = 469, which is 3 × 13 × 12 + 1 — each new layer wider than the last. Odd base, odd cube: 2197 stays odd, as 13 is.