A calculus of the absurd

19.2.2 The properties which make modular arithmetic

The first core property is this

Theorem 19.2.2 If \(a \equiv _m b\) and \(c \equiv _m d\), then
\(\seteqnumber{0}{19.}{31}\)
\begin{equation} a + c \equiv _m b + d \end{equation}

Proof: TODO

Theorem 19.2.3 If \(a \equiv _m b\) and \(c \equiv _m d\), then
\(\seteqnumber{0}{19.}{32}\)
\begin{equation} ac \equiv _m cd \end{equation}

Proof: TODO

Example 19.2.1 Calculate
\(\seteqnumber{0}{19.}{33}\)
\begin{equation} R_{10}(17^{17}) \end{equation}

This requires a bit of careful thinking about, but the result should be

\begin{align} 17^{17} &\equiv _{10} 7^{17} &\equiv _{10} 7 7^{16} \\ &\equiv _{10} 7 \left (7^2\right )^8 \\ &\equiv _{10} 7 \left (-1\right )^{8} \\ &\equiv _{10} 7 \end{align}