Solve the set of congruences 2x -= 1(mod 5) x -= 3(mod 4)

Question

Solve the set of congruences

2 x \equiv 1 (mod 5)

x \equiv 3 (mod 4)

Answer 1

Step 1
Here, we have to solve for congruences
2x=1 mod 5
x=3 mod 4
To solve such questions, we apply chinese remainder theorem.
Chinese Remainder Theorem- Let m, n are relatively prime integers. Then the system of simultaneous congruences

x \equiv a_{1} (mod m) x \equiv a_{2} (mod n)

has a unique solution modulo M = mxn, for any given integers

a_{1}, a_{2}

.
Step 2

2 x = 1 mod 5 \Rightarrow 2 \times 3 = 3 mod 5 \Rightarrow x = 3 mod 5

and we have x = 3 mod 4
Here, 4 and 5 are co-prime, therefore By chinese remainder theorem, it has unique solution module 20.
So, only possibilty is x=23 which satisfy the given congruences.
Step 3
Answer:
x=23

Solve the set of congruences 2x -= 1(mod 5) x -= 3(mod 4)

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