Question

Solve the following congruence. Make sure that the number you write is in the range [0,M−1] where M is the modulus of the congruence. If there is more than one solution, write the answer as a list separated by commas. If there is no answer, write N. 180x=276 (mod399)

Congruence
ANSWERED
asked 2020-10-18
Solve the following congruence. Make sure that the number you write is in the range [0,M−1] where M is the modulus of the congruence. If there is more than one solution, write the answer as a list separated by commas. If there is no answer, write N.
180x=276 (mod399)

Answers (1)

2020-10-19
180x =276(mod399)
It can be seen that ged (180.399) =3 and 3 divides 276, hence
180x =276(mod399) have an integer solution
This can be written as 180x+399y=276
Solve using Euclidian algorithm
\(\displaystyle{399}={180}\times{2}+{39}\)...(i)
\(\displaystyle{180}={39}\times{4}+{24}\)...(ii)
\(\displaystyle{39}={24}\times{1}+{15}\)...(iii)
\(\displaystyle{24}={15}\times{1}+{9}\)...(iv)
\(\displaystyle{15}={9}\times{1}+{6}\)...(v)
\(\displaystyle{9}={6}\times{1}+{3}\) (Stop here \(\displaystyle\because{\gcd{=}}{3}\))
Now
\(\displaystyle{3}={9}-{6}\times{1}\)
\(\displaystyle{3}={9}-{\left({15}-{9}\times{1}\right)}\times{1}\) (From (v))
\(\displaystyle{3}={9}\times{\left({2}\right)}-{\left({15}\right)}\times{1}\)
\(\displaystyle{3}={\left({24}-{15}\times{1}\right)}\times{\left({2}\right)}-{\left({15}\right)}\times{1}\) (From (iv))
\(\displaystyle{3}={\left({24}\right)}\times{\left({2}\right)}-{\left({15}\right)}\times{3}\)
\(\displaystyle{3}={\left({24}\right)}\times{\left({2}\right)}-{\left({39}-{24}\times{1}\right)}\times{3}\) (From (iii))
\(\displaystyle{3}={\left({24}\right)}\times{\left({5}\right)}-{\left({39}\right)}\times{3}\)
\(\displaystyle{3}={\left({180}-{39}\times{4}\right)}\times{\left({5}\right)}-{\left({39}\right)}\times{3}\) (From (ii))
\(\displaystyle{3}={\left({180}\right)}\times{\left({5}\right)}-{\left({39}\right)}\times{23}\)
\(\displaystyle{3}={\left({180}\right)}\times{\left({5}\right)}-{\left({399}-{180}\times{2}\right)}\times{23}\) (From (i))
\(\displaystyle{3}={\left({180}\right)}\times{\left({51}\right)}-{\left({399}\right)}\times{23}\)
multiply by 92
\(\displaystyle{276}={\left({180}\right)}\times{\left({4692}\right)}+{\left({399}\right)}\times{\left(-{2116}\right)}\)
Compare it with 180x + 399y = 276
Hence x = 4692
Since \(\displaystyle{4692}\notin{\left[{0},{338}\right]}\)
Hence answer is N
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