Question

Show that the system of congruences $\mathbf{x}\mathbf{\equiv }\mathbf{2}\mathbf{}\mathbf{(}\mathbf{mod}\mathbf{}\mathbf{6}\mathbf{)}\mathbf{}\mathbf{and}\mathbf{}\mathbf{x}\mathbf{\equiv }\mathbf{3}\mathbf{}\mathbf{(}\mathbf{mod}\mathbf{}\mathbf{9}\mathbf{)}$has no solutions.

Short answer

Inconsistent system

Step-by-step solution

Step 1

$\begin{array}{r}\mathbf{x}\mathbf{\equiv }\mathbf{2}\mathbf{mod}\mathbf{6}\mathbf{}\mathbf{}\mathbf{}\\ \mathbf{x}\mathbf{\equiv }\mathbf{1}\mathbf{mod}\mathbf{3}\mathbf{}\mathbf{}\mathbf{}\\ \mathbf{\Rightarrow }\mathbf{3}\mathbf{\mid }\mathbf{x}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ \mathbf{x}\mathbf{\equiv }\mathbf{3}\mathbf{mod}\mathbf{9}\mathbf{}\mathbf{}\mathbf{}\\ \mathbf{x}\mathbf{=}\mathbf{9}\mathbf{q}\mathbf{+}\mathbf{3}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ \mathbf{x}\mathbf{=}\mathbf{3}\mathbf{(}\mathbf{3}\mathbf{q}\mathbf{+}\mathbf{1}\mathbf{)}\\ \mathbf{\Rightarrow }\mathbf{3}\mathbf{\mid }\mathbf{x}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\end{array}$

Step 2

Hence, inconsistent system