What is the unit digit of (217) 413 × (819) 547 × (414) 624 × (342) 812 ?
8
Only unit digits of the bases matter, with cyclicities: 2 (cycle 4: 2,4,8,6), 4 (cycle 2: 4,6), 7 (cycle 4: 7,9,3,1), 9 (cycle 2: 9,1).
\(7^{413}\): \(413\bmod 4=1\Rightarrow 7\). \(9^{547}\): odd \(\Rightarrow 9\). \(4^{624}\): even \(\Rightarrow 6\). \(2^{812}\): \(812\bmod 4=0\Rightarrow 6\).
Unit digit of product \(=7\times 9\times 6\times 6 \equiv 3\times 6\times 6 \equiv 8\times 6 \equiv \mathbf{8}\).
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