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

Cabiolab 2020-12-03 Answered
Write the set of solutions of
x5(mod24)
x17(mod18)
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Expert Answer

komunidadO
Answered 2020-12-04 Author has 86 answers
Step 1
We have a set of congruence as
x5(mod24)
x17(mod18)
From the first equation we can write the set of solutions as x=[5]24orx=5t+24,tZ.
Now, we substitute for x in the second congruence x17(mod18), we get
5t+2417(mod18)
5t11(mod18)
Step 2
Multiplying both sides by 51mod(18)=11, we get,
t13(mod18)
So, t=[13]18ort=13+18s,sZ.
Substituting t back into x, we get x=5t+24orx=5(13+18s)+24=65+90s+24=90s+89,sZ.
Which gives us the solution x=[84]90x89(mod90)
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