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jmc

number theory intermediate

Problem

What is the remainder when

2^{19}

is divided by

7

?

Solution

Since

2^{3} \equiv 1 (mod 7)

and

a \equiv b (mod m)

implies

a^{c} \equiv b^{c} (mod m)

and

a d \equiv b d (mod m)

,

2^{19} = (2^{3})^{6} \cdot 2^{1} \equiv 1^{6} \cdot 2 \equiv 2 (mod 7) .

Final answer

2