# Chapter 1 - Vectors - 1.1 The Geometry and Algebra of Vectors - Exercises 1.1 - Page 17: 51

No solution in $\mathbb{Z}_{8}$.

#### Work Step by Step

The elements of $\;\mathbb{Z}_{8}\;$ is 0,1,2,3,4,5,6,7 So, if there is a solution for this equation, then x is one of these numbers. $If \;\;x=0\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(0)=5 \;\;\; {\color{Red}\rightarrow } \;\;\;\;\;\; 0\neq 5\\ so \;\;0 \;is \;not \;solution\\\\ If \;x=1\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(1)=5\;\; {\color{Red}\rightarrow } \;\;\; 6\neq 5\\ so\; 1\; is\; not\; solution.\\\\ If\;x=2\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(2)=5 \;\;\;\;{\color{Red}\rightarrow }\;\;\;\; 4\neq 5\\ so\; 2\; is\;not\; solution\\\\ If \;x=3\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(3)=5 \;\;\;{\color{Red}\rightarrow } \;\;\;\; 2\neq 5\\ so\; 3\; is\; not\; solution\\\\ If\; x=4\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(4)=5 \;\;\;{\color{Red}\rightarrow } \;\;\;0\neq 5\\ so\; 4 \;is\; not\; solution\\\\ If\; x=5\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(5)=5 \;\;\;{\color{Red}\rightarrow } \;\;\;2\neq 5\\ so\; 5 \;is\; not\; solution\\\\ If\; x=6\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(6)=5 \;\;\;{\color{Red}\rightarrow } \;\;\;4\neq 5\\ so\; 5 \;is\; not\; solution\\\\ If\; x=7\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6.(7)=5 \;\;\;{\color{Red}\rightarrow } \;\;\;2\neq 5\\ so\; 5 \;is\; not\; solution\\\\$ No solution in $\mathbb{Z}_{8}$.

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