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.

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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Ayesha Gomez

Suppose the required smallest positive integer is x.
Then by the given information, there are congruence equations
$x\equiv 2±od3,x\equiv 3±od5,x\equiv 2±od7$.
The congruence $x\equiv 2±od3$ 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 $x\equiv 3±od5$ 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 $x\equiv 2±od7$ 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
$x\equiv 2±od3,x\equiv 3±od5,x\equiv 2±od7$ is 23.
$⇒x=23$
Final Statement:
The smallest positive integer with the given conditions is 23.