Section 1
Systems of Linear Equations Modulo n
Solve the system$$\begin{array}{r}2 x+3 y=7 \\3 x+y=2\end{array}$$in integers modulo 11 by doing mod 11 arithmetic by hand. Check your solution.
Solve the system$$\begin{aligned}& 5 x+2 y=3 \\& 2 x+3 y=7\end{aligned}$$in integers modulo 13 by finding a rational solution by hand, then reducing the solution mod 13. Check your solution.
Evaluate$$\left(\begin{array}{ll}3 & 5 \\4 & 8\end{array}\right)^{-1} \bmod 11$$by finding$$\left(\begin{array}{ll}3 & 5 \\4 & 8\end{array}\right)^{-1}$$(with rational entries) and then reducing each entry modulo 11 .
Evaluate$$\left(\begin{array}{ll}3 & 5 \\4 & 8\end{array}\right)^{-1} \bmod 11$$by doing row reduction modulo 11 on the augmented matrix
Evaluate$$\left(\begin{array}{lll}2 & 3 & 1 \\0 & 1 & 5 \\3 & 7 & 2\end{array}\right)^{-1} \bmod 17$$by finding$$\left(\begin{array}{lll}2 & 3 & 1 \\0 & 1 & 5 \\3 & 7 & 2\end{array}\right)^{-1}$$(with rational entries) and then reducing each entry modulo 17.
Evaluate$$\left(\begin{array}{lll}2 & 3 & 1 \\0 & 1 & 5 \\3 & 7 & 2\end{array}\right)^{-1} \bmod 17$$by doing row reduction modulo 17 on the augmented matrix$$\left(\begin{array}{llllll}2 & 3 & 1 & 1 & 0 & 0 \\0 & 1 & 5 & 0 & 1 & 0 \\3 & 7 & 2 & 0 & 0 & 1\end{array}\right)$$
Solve the system$$\begin{aligned}& x+2 y+2 z=5 \\& 2 x+y+2 z=7 \\& 2 x+2 y+z=3\end{aligned}$$in integers modulo 13 by doing mod 13 arithmetic by hand. Check your solution.
Solve the system$$\begin{array}{r}5 x+2 y+z=6 \\2 x+3 y+4 z=1 \\4 x+y+3 z=2\end{array}$$in integers modulo 7 by finding a rational solution by hand, then reducing the solution mod 7. Check your solution.
Use a computer algebra system to solve the system$$\begin{array}{r}w+x+y+z=23 \\w+2 x+3 y+4 z=19 \\w+4 x+9 y+16 z=42 \\w+8 x+27 y+64 z=73\end{array}$$in integers modulo 91 by solving the system in $\mathbb{Q}$ and reducing modulo 91 .
Use a computer algebra system to solve the system$$\begin{array}{r}w+x+y+z=23 \\w+2 x+3 y+4 z=19 \\w+4 x+9 y+16 z=42 \\w+8 x+27 y+64 z=73\end{array}$$in integers modulo 91 by solving the corresponding matrix equation modulo 91 .