Write the set of solutions of x -= 5 (mod 24) x -= 17(mod 18)

Question

Write the set of solutions of

x \equiv 5 (mod 24)

x \equiv 17 (mod 18)

Answer 1

Step 1
We have a set of congruence as

x \equiv 5 (mod 24)

x \equiv 17 (mod 18)

From the first equation we can write the set of solutions as

x = {[5]}_{24} or x = 5 t + 24, t \in Z

.
Now, we substitute for x in the second congruence

x \equiv 17 (mod 18)

, we get

5 t + 24 \equiv 17 (mod 18)

\Rightarrow 5 t \equiv 11 (mod 18)

Step 2
Multiplying both sides by

5^{- 1} mod (18) = 11

, we get,

t \equiv 13 (mod 18)

So,

t = {[13]}_{18} or t = 13 + 18 s, s \in Z

.
Substituting t back into x, we get

x = 5 t + 24 or x = 5 (13 + 18 s) + 24 = 65 + 90 s + 24 = 90 s + 89, s \in Z

.
Which gives us the solution

x = {[84]}_{90} \Rightarrow x \equiv 89 (mod 90)

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