# Find the remainder when $2(26!)$ is divided by $29$.

Find the remainder when $2(26!)$ is divided by $29$.

So I know I'm going to use Wilson's theorem and then I would have $28!=-1(\mod29\:)$ but what is the next step? Step by Step explanation please!

• If $28! \equiv -1 \pmod{29}$, what is $27!$ congruent to. And what $26!$? – Daniel Fischer May 12 '14 at 22:20
• 27!≡-1 (mod 28), 26!≡-1(mod27) correct? – Lil May 12 '14 at 22:23
• – lhf May 12 '14 at 22:26

$\begin{eqnarray} {\bf Hint}\ \ \ {\rm mod}\ 29\!:\,\ {-}1\! \overset{\rm Wilson}\equiv 28!\, \equiv &&\ (\color{#c00}{28})\ (\color{#0a0}{27})26!\\ \equiv&& (\color{#c00}{-1})(\color{#0a0}{-2}) 26!\\ \equiv &&\qquad\ \ \,2\cdot 26!\end{eqnarray}$

• where did the -1 and -2 come from? is it 28-29 and 27-29? – Lil May 12 '14 at 22:26
• @Lil mod $\,29\!:\ 28\equiv -1\,$ by $\,20\mid 28-(-1)\$ and $\ 27\equiv -2\,$ by $\,29\mid 27-(-2)\ \$ – Bill Dubuque May 12 '14 at 22:27
• ok after that would I multiply (-1)(-2) and get 2(26!)≡−1mod29 – Lil May 12 '14 at 22:29
• @Lil Exactly, just as I wrote above. – Bill Dubuque May 12 '14 at 22:30

$$28!\equiv -1\bmod 29$$ or $$28\cdot27 \cdot 26!\equiv -1\bmod 29$$ which is $$2\cdot 14\cdot27 \cdot 26!\equiv -1\bmod 29$$ Now let $a$ and $b$ be modular inverses of $14,27$ respectively.(such $a,b$ exist because $gcd(14,29)=1$ and $gcd(27,29)=1$).

Multiply both sides by $a\cdot b$. The result is what you want.

• how did you change 28 to 2, 27 to 14? – Lil May 12 '14 at 22:24
• $28=2\cdot 14$. – Fermat May 12 '14 at 22:26