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

Question

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

Sadie Eaton · Accepted Answer

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 \equiv a (m o d m)

x \equiv b (m o d n)

Furthermore, any two solutions x and y are congruent modulo mn.
2) Theorem: Addition and Multiplication Properties:
If

a \equiv b (m o d n)

and x is any integer, then

a + x - = b + x (m o d n) a n d a x \equiv b x (m o d n)

.
Explanation:
Consider the system of congruences

x \equiv (m o d 5)

x \equiv 3 (m o d 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 + 5 k

for some integer k and substitute this expression for x into the second congruence.

2 + 5 k \equiv 3 (m o d 8)

By using addition property,

2 + 5 k + (— 2) \equiv 3 + (— 2) (m o d 8)

= 5 k = 1 (m o d 8)

By using multiplication property,

\Rightarrow 2 + 5 k + (- 2) \equiv 3 + (- 2) (m o d 8)

\Rightarrow 5 k \equiv 1 (m o d 8)

By using multiplication property,

5 * 5 k - \equiv 5 * 1 (m o d 8)

\Rightarrow 25 k \equiv 5 (m o d 8)

Since

25 \equiv 1 (m o d 8)

,

\Rightarrow k \equiv 5 (m o d 8)

Thus,

x = 2 + 5 (5) = 27

satisfies the system and

x \equiv 27 (m o d 5 * 8) o r x \equiv 27 (m o d 40)

gives all solutions to the given system of congruences.

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

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