x &#x2261;<!-- ≡ --> 11 <mspace width="1em" /> ( mod <mspace width="0.333em" /> 84

misurrosne 2022-06-27 Answered
x 11 ( mod 84 )
x 23 ( mod 36 )
I have the bulk of the work done for this;
x = 11 + 84 j
x = 23 + 36 k
11 + 84 j 23 ( mod 36 )
84 j 12 ( mod 36 )
12 j 12 ( mod 36 )
j 1 ( mod 36 )
j = 1 + 36 n
Thus this system is true for any x of the form
x = 11 + 84 ( 1 + 36 n ) = 95 + 3024 n
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Answers (2)

Harold Cantrell
Answered 2022-06-28 Author has 21 answers
Correct is:
  12 j 12 ( mod 36 ) 36 12 ( j 1 ) c a n c e l   12 3 j 1 j = 1 + 3 n .
Therefore   x = 11 + 84 j = 11 + 84 ( 1 + 3 n ) = 95 + 252 n , as claimed.

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Ayanna Trujillo
Answered 2022-06-29 Author has 13 answers
x 11 ( mod 84 ), so x = 84 k + 11, and x 23 ( mod 36 ), so x = 36 n + 23. So 84 k + 11 = 36 n + 23, and 84 k 36 n = 12. So 7 k 3 n = 1. So 3 n = 7 k 1, and n = 7 k 1 3 = 2 k + k 1 3 . Thus 3 | k 1, and k = 3 t + 1, and n = 2 ( 3 t + 1 ) + t = 7 t + 2. So x = 36 n + 23 = 36 ( 7 t + 2 ) + 23 = 252 t + 95 with t Z

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