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

Marvin Mccormick

Marvin Mccormick

Answered question


Solve the following systems of congruences.
x2(mod 5)

Answer & Explanation

Sadie Eaton

Sadie Eaton

Skilled2020-10-29Added 104 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
xa(mod m)
xb(mod n)
Furthermore, any two solutions x and y are congruent modulo mn.
2) Theorem: Addition and Multiplication Properties:
If ab(mod n) and x is any integer, then a+x=b+x(mod n) and axbx(mod n).
Consider the system of congruences x(mod 5)
x3(mod 8)
Since 5 and 8 are relatively prime, then (5,8)=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.
2+5k3(mod 8)
By using addition property,
2+5k+(2)3+(2)(mod 8)
=5k=1(mod 8)
By using multiplication property,
2+5k+(2)3+(2)(mod 8)
5k1(mod 8)
By using multiplication property,
55k51(mod 8)
25k5(mod 8)
Since 251(mod 8),
k5(mod 8)
Thus, x=2+5(5)=27 satisfies the system and x27(mod 58) or x27(mod 40) gives all solutions to the given system of congruences.

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