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jmc

number theory intermediate

Problem

Find the remainder of

7^{17}

when it is divided by 5.

Solution

7^{4} \equiv 2^{4} = 16 \equiv 1 (mod 5)

, so

7^{17} = 7^{4 \cdot 4 + 1} = (7^{4})^{4} \cdot 7^{1} \equiv 1^{4} \cdot 2 \equiv 2 (mod 5)

.

Final answer

2