 # Solve the following systems of congruences. xequiv 4(mod 7) Khadija Wells 2021-03-06 Answered
Solve the following systems of congruences.
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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

By using multiplication property,

Since ,

Therefore, the system of congruences is

Since 7 and 8 are relatively prime then $\left(7,8\right)=1$.
Then, by using theorem there exists an integer x that satisfies the system of congruences.
From the first congruence $x=4+7k$ for some integer k and substitute this expression for x into the second congruence.

Since ,

Since (7, 8) = I then by using cancellation law,

Thus, $x=4+7\left(1\right)=11$ satisfies the system and gives all solutions to the given system of congruences.