Let x = (633)24 – (277)38+ (266)54. What is the units digit of x?
8
The units digit of each power depends only on the base's last digit and the exponent modulo its cycle.
(633)24: Powers of 3 cycle 3, 9, 7, 1 (length 4). \(24\bmod 4=0\), so units digit = 1.
(277)38: Powers of 7 cycle 7, 9, 3, 1. \(38\bmod 4=2\), so units digit = 9.
(266)54: Any power of a number ending in 6 ends in 6.
Combining: units digit of \(1-9+6\) = units digit of \(11-9+6=8\). Hence the answer is 8.
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