Find the remainder when $13^{13}$ is divided by $25$.

Question

Find the remainder when $13^{13}$ is divided by $25$.

Here is my attempt, which I think is too tedious:

Since $13^{2} \equiv 19 (\text{mod} \ 25),$ we have $13^{4} \equiv 19^{2} \equiv 11 (\text{mod} \ 25)$ and $13^{8} \equiv 121 \equiv 21 (\text{mod} \ 25).$ Finally, we have $13^{8+4} \equiv 13^{12} \equiv 21\times 11 \equiv 231 \equiv 6 (\text{mod} \ 25)$ and hence $13^{13} \equiv 3 (\text{mod} \ 25).$

Is there a less tedious way to find the remainder? Thank you.

Answer 1

Not really, you could only use this $$19\equiv -6\pmod {25}$$ and $$121\equiv -4\pmod {25}$$ That would perhaps save you some time.

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By Euler theorem you could also see that
$$ 13^{20}\equiv1\pmod{25} \implies 13^{10}\equiv \pm1\pmod{25} $$ so

Answer 2

$13$ is coprime with $25$, so Euler-Fermat tells us that $$ 13^{\varphi(25)}\equiv1\pmod{25} $$ Since $\varphi(25)=20$, this is not much of a help.

You can try with repeated squares: $13=1+4+8$; since $$ 13^2\equiv 19,\quad 13^4\equiv 19^2\equiv11\quad 13^8\equiv 11^2\equiv21 $$ we get $$ 13^{13}\equiv 13\cdot 11\cdot 21\equiv3\pmod{25} $$ which actually is what you did. I don't think there's much easier methods.

Answer 3

$\!\bmod 25\!:\ 13\equiv \overbrace{2^{\large -1}\!\equiv 3^{\large -3}}^{\Large 2\ \, \equiv\,\ 3^{\LARGE 3}}$ $\Rightarrow 13^{\large 13}\!\equiv 3^{\large -39}\!\equiv 3,\ $ by $\ 3^{\large 40}\!\equiv\!\!\! \overbrace{(3^{\large \color{#c00}{20}})^{\large 2}\!\equiv 1}^{\large\quad\ \ \color{#c00}{20}\ =\ \phi(25)}$

Answer 4

We have

• $2\cdot 13 = 26 \equiv 1 \mod 25$
• $\Rightarrow 13^{13} \cdot \color{blue}{2^{13}} \equiv 1 \mod 25$
• $\color{blue}{2^{13}} \equiv 2^{10}\cdot 2^3 \equiv -8 \color{blue}{\equiv 17 \mod 25}$ $$\Rightarrow \boxed{13^{13} \equiv 17^{-1} \equiv 3 \mod 25}$$