Given information:

The congruences are \(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)},{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\).

Consider the given congruences

\(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)},{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\)

The congruences \(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)}\) means if x is divided by 6, the remainder is 1.

So number x is one of the numbers in the following list:

7,13,19,25,31,37,43,...

Similarity, the congruence \(\displaystyle{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\) means if x is divided by 8, the remainder is 7.

So the number x is one of the numbers in the following list:

7,15,22,29,36,43,49,56,63,...

The smallest number that is found in both the lists is 7, so the smallest number that solves the congruences

\(\displaystyle{x}\equiv{3}{\left({b}\text{mod}{7}\right)},{x}\equiv{4}{\left({b}\text{mod}{5}\right)}\) is 7.

\(\displaystyle\Rightarrow{x}={7}\).

Final Statement:

The smallest positive integer that solves the congruences

\(\displaystyle{x}\equiv{3}{\left({b}\text{mod}{7}\right)},{x}\equiv{4}{\left({b}\text{mod}{5}\right)}\) is x= 7.

The congruences are \(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)},{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\).

Consider the given congruences

\(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)},{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\)

The congruences \(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)}\) means if x is divided by 6, the remainder is 1.

So number x is one of the numbers in the following list:

7,13,19,25,31,37,43,...

Similarity, the congruence \(\displaystyle{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\) means if x is divided by 8, the remainder is 7.

So the number x is one of the numbers in the following list:

7,15,22,29,36,43,49,56,63,...

The smallest number that is found in both the lists is 7, so the smallest number that solves the congruences

\(\displaystyle{x}\equiv{3}{\left({b}\text{mod}{7}\right)},{x}\equiv{4}{\left({b}\text{mod}{5}\right)}\) is 7.

\(\displaystyle\Rightarrow{x}={7}\).

Final Statement:

The smallest positive integer that solves the congruences

\(\displaystyle{x}\equiv{3}{\left({b}\text{mod}{7}\right)},{x}\equiv{4}{\left({b}\text{mod}{5}\right)}\) is x= 7.