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

naivlingr 2021-03-07 Answered
Solve the set of congruences
2x1(mod5)
x3(mod4)
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Expert Answer

smallq9
Answered 2021-03-08 Author has 106 answers
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 xa1(modm)xa2(modn) has a unique solution modulo M = mxn, for any given integers a1,a2.
Step 2
2x=1mod52×3=3mod5x=3mod5
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
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