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Math Help - Find all integers x that satisfy the following system of congruences:

  1. #1
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    Find all integers x that satisfy the following system of congruences:

    Find all integers x that satisfy the following system of congruences:
    9x≡1(mod 20) ; 52x≡2 (mod 209) ; 8x≡(mod 21)


    need elaborate and quick answers please
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  2. #2
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    Re: Find all integers x that satisfy the following system of congruences:

    Hello, virebbala90!

    There is an omission . . .


    Find all integers x that satisfy the following system of congruences:

    . . \begin{array}{cccc}9x &\equiv& 1& \text{(mod 20)} \\ 52x &\equiv& 2 & \text{(mod 209)} \\ 8x &\equiv& {\color{red}?} & \text{(mod 21)} \end{array}
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  3. #3
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    Re: Find all integers x that satisfy the following system of congruences:

    First, find the prime factorization for each: 20 = 2^2\cdot 5, 209 = 11\cdot 19, 21 = 3\cdot 7 Then, consider 9x mod 4 and mod 5 (this will allow you to easily calculate x mod 4 and mod 5). Then consider 52x mod 11 and mod 19 to calculate x mod 11 and mod 19. Finally, consider 8x mod 3 and mod 7 to calculate x mod 3 and mod 7. Once you know x mod 4, 5, 11, 19, 3, and 7, use the Chinese Remainder Theorem to find x mod 4\cdot 5\cdot 11\cdot 19\cdot 3\cdot 7 = 87780.
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