How does the ×11 trick extend to longer numbers, e.g. 253 × 11?

Keep the first and last digits; between them, write the sum of each adjacent pair — carrying as needed.

253 × 11 → first digit 2, then (2+5)=7, then (5+3)=8, then last digit 3 → 2783.

Check: 253 × 11 = 2530 + 253 = 2783. ✓

With carries it still works, right to left: 758 × 11 → 7 | (7+5)=12 | (5+8)=13 | 8 → resolve carries → 8338. (Check: 7580 + 758 = 8338. ✓)