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

ringearV

ringearV

Answered question

2021-02-21

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

Answer & Explanation

un4t5o4v

un4t5o4v

Skilled2021-02-22Added 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.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school geometry

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?