 # Solve the set of congruences 2x -= 1(mod 5) x -= 3(mod 4) naivlingr 2021-03-07 Answered
Solve the set of congruences
$2x\equiv 1\left(\text{mod}5\right)$
$x\equiv 3\left(\text{mod}4\right)$
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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 $x\equiv {a}_{1}\left(\text{mod}m\right)x\equiv {a}_{2}\left(\text{mod}n\right)$ has a unique solution modulo M = mxn, for any given integers ${a}_{1},{a}_{2}$.
Step 2
$2x=1\text{mod}5⇒2×3=3\text{mod}5⇒x=3\text{mod}5$
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
x=23