If x = (164) 169 + (333) 337 – (727) 726 , then what is the units digit of x?
8
The units digit depends only on the base's units digit and the exponent.
Term 1 — \(164^{169}\): Powers of 4 cycle as 4, 6 (cyclicity 2). Exponent 169 is odd, so units digit is 4.
Term 2 — \(333^{337}\): Powers of 3 cycle as 3, 9, 7, 1 (cyclicity 4). \(337 \bmod 4 = 1\), so units digit is 3.
Term 3 — \(727^{726}\): Powers of 7 cycle as 7, 9, 3, 1 (cyclicity 4). \(726 \bmod 4 = 2\), so units digit is 9.
Combine: units digit of \(x = 4 + 3 - 9 = -2 \equiv 8 \pmod{10}\). Hence the units digit is 8.
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