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.

DofotheroU 2021-05-06 Answered
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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Expert Answer

Ayesha Gomez
Answered 2021-05-08 Author has 104 answers

Suppose the required smallest positive integer is x.
Then by the given information, there are congruence equations
x2±od3,x3±od5,x2±od7.
The congruence x2±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 x3±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 x2±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
x2±od3,x3±od5,x2±od7 is 23.
x=23
Final Statement:
The smallest positive integer with the given conditions is 23.

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