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

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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Neelam Wainwright
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.