ringearV

2021-02-21

Solve the following systems of congruences.

un4t5o4v

Expert

Formula used:
1) Theorem: System of congruences:
Let m and n be relatively prime and a and b integers. There exists an integer x that satisfies the system of congruences

Furthermore, any two solutions x and y are congruent modulo mn.
2) Theorem: Addition and Multiplication Properties:
If and x is any integer, then .
3) Theorem: Cancellation Law:
If .
Explanation:
Consider the system of congruences

Since 5 and 3 are relatively prime, then $\left(5,3\right)=1$.
Then, by using theorem there exists an integer x that satisfies the system of congruences.
From the first congruence $x=4+5k$ for some integer k and substitute this expression for x into the second congruence.

By using addition property,

Since ,

Since $\left(2,3\right)=1$ then by using cancellation law,

Now, ,
Therefore,
Thus, $x=4+5\left(2\right)=14$ satisfies the system and or gives all solutions to the given system of congruences.

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