# To find: The smallest positive integer such that if we divide

To find:
The smallest positive integer such that if we divide it by three, the remainder is 2; if we divide it by five, the remainder is 3; if we divide it by seven, the remainder is 2.

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Tuthornt
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.