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Number Theory Level 3

What is the unit digit of $333^1 \times 333^2 \times 333^3\times\cdots \times 333^{100}$ ?

1 3 7 9

1 solution

J C
Apr 6, 2016

$333^1\cdot 333^2\cdot 333^3...333^{100}$ is equal to $333^{5050}$ . We get 5050 from the sum of 1 to 100. Since the cycle of last digits goes 3, 9, 7, 1 for a power of 3, 5050 is 2 in mod 4, thus the second number in the pattern, or 9.

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