Given information:

The congruences \(\displaystyle{x}\equiv{3}{\left({b}\ mod \ {7}\right)},{x}\equiv{4}{\left({b}\ mod \ {5}\right)}\)

Consider the given congruences

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

‘The congruence \(\displaystyle{x}\equiv{3}{\left({b}\ mod \ {7}\right)}\) means if x is divided by 7, the remainder is 3.

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

3, 10, 17, 24, 31, 38, 45, -

Similarly, the congruence \(\displaystyle{x}\equiv{4}{\left({b}\ mod \ {5}\right)}\) means if x is divided by 5, the remainder is 4.

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

4,9, 14, 19,24, 29,34, 39, 44, -

The smallest number that is found in both the lists is 24, so the

smallest number that solves the congruences

\(\displaystyle{x}\equiv{3}{\left({b}\ mod \ {7}\right)},{x}\equiv{4}{\left({b}\ mod \ {5}\right)}\) is 24.

\(\displaystyle\Rightarrow{x}={24}\)

Final Statement:

The smallest positive integer that solves the congruences

\(\displaystyle{x}\equiv{3}{\left({b}\ mod \ {7}\right)},{x}\equiv{4}{\left({b}\ mod \ {5}\right)}\) is x = 24.