Solve the following systems of congruences. x≡4(mod 5) x≡3(mod 8) x≡2(mod 3)

ringearV

ringearV

Answered

2021-02-21

Solve the following systems of congruences.
x4(mod 5)
x3(mod 8)
x2(mod 3)

Answer & Explanation

un4t5o4v

un4t5o4v

Expert

2021-02-22Added 105 answers

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
x(mod m)
xb(mod n)
Furthermore, any two solutions x and y are congruent modulo mn.
2) Theorem: Addition and Multiplication Properties:
If a=b(mod n) and x is any integer, then a+xb+x(mod n) and axbx(mod n).
3) Theorem: Cancellation Law:
If axay(mod n) and (a,n)=1, then xy(mod n).
Explanation:
Consider the system of congruences
x4(mod 5)
x2(mod 3)
Since 5 and 3 are relatively prime, then (5,3)=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.
4+5k2(mod 3)
By using addition property,
4+5k+(4)2+(4)(mod 3)
5k=2(mod 3)
Since 52(mod 3),
2k2(mod 3)
Since (2,3)=1 then by using cancellation law,
k1(mod 3)
Now, 12(mod 3),
Therefore, k=2(mod 3)
Thus, x=4+5(2)=14 satisfies the system and x=14(mod 53) or x=14(mod 15) gives all solutions to the given system of congruences.

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