Solve the following systems of congruences. x≡4(mod 5) x≡3(mod 8) x≡2(mod 3)
Solve the following systems of congruences.
Answer & Explanation
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:
Consider the system of congruences
Since 5 and 3 are relatively prime, then .
Then, by using theorem there exists an integer x that satisfies the system of congruences.
From the first congruence for some integer k and substitute this expression for x into the second congruence.
By using addition property,
Since then by using cancellation law,
Thus, satisfies the system and or gives all solutions to the given system of congruences.