Marvin Mccormick

2020-10-28

Solve the following systems of congruences.

### Answer & Explanation

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 .
Explanation:
Consider the system of congruences

Since 5 and 8 are relatively prime, then $\left(5,8\right)=1$.
Then, by using a theorem, there exists an integer x that satisfies the system of congruences.
From the first congruence $x=2+5k$ for some integer k and substitute this expression for x into the second congruence.

By using addition property,

By using multiplication property,

By using multiplication property,

Since ,

Thus, $x=2+5\left(5\right)=27$ satisfies the system and gives all solutions to the given system of congruences.

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