remolatg

2021-02-18

Solve the following systems of congruences.

Aamina Herring

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,
Since ,

By using multiplication property,

Since ,

Now, consider

By using multiplication property,

Since ,

Therefore, the system of congruences is

Since 3 and 7 are relatively prime then $\left(3,7\right)=1$.
From the first congruence $x=1+3k$ for some integer k and substitute this expression for x into the second congruence.

By using multiplication property,

Since

Since $\left(2,7\right)=1$ then by using cancellation law,

Thus, $x=1+3\left(3\right)=10$ satisfies the system and gives all solutions to the given systems of congruences.

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