Modular Arithmetic $-17 \cdot 23 \mod 11$

Question

Evaluate $-17 \cdot 23 \mod 11$.

Can you show how to calculate this? I'm trying to understand how you do this, so clear instructions would be great. Thank you.

Answer 1

If you are doing a calculation $\mod{11}$, you want to find the remainder when the answer is divided by $11$. The naive way to do this problem is to multiply $-17$ and $23$, then find the remainder when the product is divided by $11$.

\begin{align*} -17 \cdot 23 \pmod{11} & \equiv -391 \pmod{11}\\ & \equiv 5 \pmod{11} \end{align*}

since $$-391 = -36 \cdot 11 + 5$$ has remainder $5$ when divided by $11$.

However, you can simplify by the calculations by reducing $\mod 11$ before you multiply since if $a \equiv a' \pmod{n}$ and $b \equiv b' \pmod{11}$, then

$$ab \equiv a'b' \pmod{n}$$

Since $$-17 = -2 \cdot 11 + 5$$ we obtain $-17 \equiv 5 \pmod{11}$. Since $$23 = 2 \cdot 11 + 1$$ we obtain $$23 \equiv 1 \pmod{11}$$

Thus, using the property cited above,

\begin{align*} -17 \cdot 23 \pmod{11} & \equiv 5 \cdot 1 \pmod{11}\\ & \equiv 5 \pmod{11} \end{align*} which agrees with the previous result. However, the calculations are much simpler if you reduce $\mod{11}$ first.