Illustration of Theorem 5.5

$$12\equiv 22(\mathrm{mod}10),2\mid 10,2\mid 12,2\mid 22.$$

Illustration of Theorem 5.6

$$12\equiv 42(\mathrm{mod}30),(12,30)=(42,30)=6.$$

Illustration of Theorem 5.7

$$3\equiv 13\equiv 23\equiv 33\equiv 43(\mathrm{mod}10).$$

The difference between any two distinct integers above is $\ge 10$.

Illustration of Theorem 5.8

$$13\equiv 23\equiv 3(\mathrm{mod}10).$$

Illustration of Theorem 5.9

$$4\equiv 19(\mathrm{mod}3),4\equiv 19(\mathrm{mod}5),4\equiv 19(\mathrm{mod}15).$$

$$4\equiv 10(\mathrm{mod}3),4\equiv 10(\mathrm{mod}6),4\not\equiv 10(\mathrm{mod}18).$$