How to prove that the remainder of $15^{2098}$ divided by $14$ is $1$?

Question

I have to compute the remainder of $15^{2098}$ divided by $14$, I know that the answer will be $1$ because $15$ is $14+1$ and the remainder of every $15$'s power will be $1$ when divided by $14$, the problem is that I can't write that as answer because it is not too mathematical, so I want know how can I prove this without having to do all the long computation.

score 5 · Accepted Answer · answered Nov 08 '20 at 16:58

5

$$15\equiv 1 \pmod{14}$$

$$15^{2098}\equiv 1^{2098}\equiv 1 \pmod{14}$$

answered Nov 08 '20 at 16:58

Lion Heart

7,073

score 1 · Answer 2 · answered Nov 08 '20 at 17:14

1

If you have enough information about congruences, then use the answer by Lion Heart. Otherwise, notice that $(1+14)^{2098}$, when multiplied out, gives 1 followed by a lot of further terms all of which have $14$ as a factor.

answered Nov 08 '20 at 17:14

Andreas Blass

71,833

How to prove that the remainder of $15^{2098}$ divided by $14$ is $1$?

2 Answers2