Simultaneous linear congruences

3

1. Solve \(7 x \equiv 6 ( \bmod 19 )\).
2. Show that the following simultaneous linear congruences have no solution. $$x \equiv 3 ( \bmod 4 ) , x \equiv 4 ( \bmod 6 )$$

3 Determine all integers \(x\) for which \(x \equiv 1 ( \bmod 7 )\) and \(x \equiv 22 ( \bmod 37 )\) and \(x \equiv 7 ( \bmod 67 )\).
Give your answer in the form \(\mathrm { x } = \mathrm { qn } + \mathrm { r }\) for integers \(n , q , r\) with \(q > 0\) and \(0 \leqslant \mathrm { r } < \mathrm { q }\).

4 Solve the simultaneous linear congruences \(x \equiv 1 ( \bmod 3 ) , x \equiv 5 ( \bmod 11 ) , 2 x \equiv 5 ( \bmod 17 )\).

1 Determine the solution of the simultaneous linear congruences $$x \equiv 4 ( \bmod 7 ) , \quad x \equiv 25 ( \bmod 41 ) .$$