Cube of 7

7 cubed equals 343. Cubes show up in volume calculations — a cube with side 7 holds 343 cubic units.

Cubes carry the fingerprint of their base. The final digit of 343 is 3 — always the cube of the base's own final digit. Divided by 9, a cube only ever leaves 0, 1 or 8, and 343 leaves 1. The volume picture explains the name: a 7 × 7 floor of unit cubes holds 49 of them, and 7 such layers stack to 343 — the shape itself is the exponent at work. The gaps grow like a parabola: 343 − 216 = 127, which is 3 × 7 × 6 + 1 — each new layer wider than the last. Odd base, odd cube: 343 stays odd, as 7 is.