Suppose the required smallest positive integer is x.

Then by the given information, there are congruence equations

\(\displaystyle{x}\equiv{2}\pm{o}{d}{3},{x}\equiv{3}\pm{o}{d}{5},{x}\equiv{2}\pm{o}{d}{7}\).

The congruence \(\displaystyle{x}\equiv{2}\pm{o}{d}{3}\) means if x is divided by 3, the remainder is 2.

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

2,5, 8, 11, 14, 17, 20, 23, 26,29,...

Similarly, the congruence \(\displaystyle{x}\equiv{3}\pm{o}{d}{5}\) means if x is divided by 5, the remainder is 3.

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

3,8, 13, 18, 23, 28, 33, 38,43,...

The congruence \(\displaystyle{x}\equiv{2}\pm{o}{d}{7}\) means if x is divided by 7, the remainder is 2.

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

2,9, 16,23, 30, 37,44,...

The smallest number that is found in the above three lists is 23.

So the smallest number that solves the congruences

\(\displaystyle{x}\equiv{2}\pm{o}{d}{3},{x}\equiv{3}\pm{o}{d}{5},{x}\equiv{2}\pm{o}{d}{7}\) is 23.

\(\displaystyle\Rightarrow{x}={23}\)

Final Statement:

The smallest positive integer with the given conditions is 23.