Let x = (733)24 - (397)38 + (566)54. What is the units digit of x?

CONCEPT

The units digit of a power depends only on the units digit of its base. For digits such as 3 and 7, the units digits repeat in a cycle of length 4.

For sums and differences, reduce each term to its units digit, combine the signed residues, and then read the final result modulo 10.

APPLICATION

1. The units digit of (733)24 is the units digit of 324. The cycle for 3 is 3, 9, 7, 1, and 24 is divisible by 4, so this term contributes 1.

2. The units digit of (397)38 is the units digit of 738. The cycle for 7 is 7, 9, 3, 1, and 38 leaves remainder 2 when divided by 4, so this term contributes 9.

3. The units digit of (566)54 is 6, because every positive power of a number ending in 6 also ends in 6.

4. Now combine the signed unit residues: 1 - 9 + 6 = -2. Modulo 10, -2 has the same units digit as 8.

CROSS-CHECK

Adding 10 to the negative residue gives -2 + 10 = 8, so the units digit is stable as 8.

Result: the units digit of x is 8.