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

banganX 2021-02-21 Answered

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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Expert Answer

opsadnojD
Answered 2021-02-23 Author has 9667 answers

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.

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