# To find: The smallest positive integer that solves the congruencesx\equiv 3(\bmod 7), x\equiv 4(\bmod 5)

To find: The smallest positive integer that solves the congruences
$$\displaystyle{x}\equiv{3}{\left({b} \ mod \ {7}\right)},{x}\equiv{4}{\left({b}\ mod \ {5}\right)}$$

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Given information:
The congruences $$\displaystyle{x}\equiv{3}{\left({b}\ mod \ {7}\right)},{x}\equiv{4}{\left({b}\ mod \ {5}\right)}$$
Consider the given congruences
$$\displaystyle{x}\equiv{3}{\left({b}\ mod \ {7}\right)},{x}\equiv{4}{\left({b}\ mod \ {5}\right)}$$
‘The congruence $$\displaystyle{x}\equiv{3}{\left({b}\ mod \ {7}\right)}$$ means if x is divided by 7, the remainder is 3.
So the number x is one of the numbers in the following list:
3, 10, 17, 24, 31, 38, 45, -
Similarly, the congruence $$\displaystyle{x}\equiv{4}{\left({b}\ mod \ {5}\right)}$$ means if x is divided by 5, the remainder is 4.
So the number x is one of the numbers in the following list:
4,9, 14, 19,24, 29,34, 39, 44, -
The smallest number that is found in both the lists is 24, so the
smallest number that solves the congruences
$$\displaystyle{x}\equiv{3}{\left({b}\ mod \ {7}\right)},{x}\equiv{4}{\left({b}\ mod \ {5}\right)}$$ is 24.
$$\displaystyle\Rightarrow{x}={24}$$
Final Statement:
The smallest positive integer that solves the congruences
$$\displaystyle{x}\equiv{3}{\left({b}\ mod \ {7}\right)},{x}\equiv{4}{\left({b}\ mod \ {5}\right)}$$ is x = 24.