What is the unit's digit of 213 x 212 x 347 x 468 x 778?

Concept: The unit's digit of a product of several numbers depends only on the unit's digits of the individual factors -- every other digit can be ignored. To find it, multiply the unit's digits of the factors together, keeping only the unit's digit of the running product at each step.

1. Unit's digits of the factors: 213 -> 3, 212 -> 2, 347 -> 7, 468 -> 8, 778 -> 8.

2. 3 x 2 = 6, so the running unit's digit is 6.

3. 6 x 7 = 42, so the running unit's digit is 2.

4. 2 x 8 = 16, so the running unit's digit is 6.

5. 6 x 8 = 48, so the running unit's digit is 8.

Cross-check: the exact product 213 x 212 x 347 x 468 x 778 = 5,705,193,637,728, whose last digit is 8, confirming the stepwise calculation.

So the unit's digit of the product is 8.