43256 × 57343 — what is the units digit of the product?

Concept: The units digit of a product depends only on the units digits of its factors, and the units digit of any integer power an repeats in a short cycle (length at most 4). To find the units digit of an, find a's units-digit cycle and use the entry at position (n mod cycle length), mapping a remainder of 0 to the cycle's last position.

1. Reduce each base to its units digit: 43 ends in 3; 57 ends in 7.

2. 3's units-digit cycle is 3, 9, 7, 1 (length 4). Since 256 mod 4 = 0, take the 4th (last) entry: 3256 ends in 1.

3. 7's units-digit cycle is 7, 9, 3, 1 (length 4). Since 343 mod 4 = 3, take the 3rd entry: 7343 ends in 3.

4. Multiply the two units digits found: 1 × 3 = 3, so 43256 × 57343 ends in 3.

Cross-check: 34 ≡ 1 (mod 10), and 256 is a multiple of 4, so 3256 ≡ 1 (mod 10) — confirming step 2 independently. Similarly 74 ≡ 1 (mod 10), and 343 = 4×85 + 3, so 7343 ≡ 73 (mod 10) = 3 — confirming step 3.

Answer: 3.