To find: The smallest positive integer that solves the congruences x\equiv 1(\bmod 6), x\equiv 7(\bmod 8)

geduiwelh 2021-05-08 Answered
To find: The smallest positive integer that solves the congruences
\(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)},{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\)

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Expert Answer

Neelam Wainwright
Answered 2021-05-10 Author has 7115 answers
Given information:
The congruences are \(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)},{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\).
Consider the given congruences
\(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)},{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\)
The congruences \(\displaystyle{x}\equiv{1}{\left({b}\text{mod}{6}\right)}\) means if x is divided by 6, the remainder is 1.
So number x is one of the numbers in the following list:
7,13,19,25,31,37,43,...
Similarity, the congruence \(\displaystyle{x}\equiv{7}{\left({b}\text{mod}{8}\right)}\) means if x is divided by 8, the remainder is 7.
So the number x is one of the numbers in the following list:
7,15,22,29,36,43,49,56,63,...
The smallest number that is found in both the lists is 7, so the smallest number that solves the congruences
\(\displaystyle{x}\equiv{3}{\left({b}\text{mod}{7}\right)},{x}\equiv{4}{\left({b}\text{mod}{5}\right)}\) is 7.
\(\displaystyle\Rightarrow{x}={7}\).
Final Statement:
The smallest positive integer that solves the congruences
\(\displaystyle{x}\equiv{3}{\left({b}\text{mod}{7}\right)},{x}\equiv{4}{\left({b}\text{mod}{5}\right)}\) is x= 7.
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