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Suppose that x is an integer that satisfies the following congruences

\(:\begin{align*} 3+x &\equiv 2^2 \pmod{3^3} \\ 5+x &\equiv 3^2 \pmod{5^3} \\ 7+x &\equiv 5^2 \pmod{7^3} \end{align*}\)

What is the remainder when x is divided by 105?

 Dec 16, 2020
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(3 + x) mod 3^3 = 2^2
(5 + x) mod 5^3 = 3^2
(7 + x) mod 7^3 = 5^2, solve for x

 

Using CRT + MMI, we have:

 

x = 1157625 n + 506629, where n =0, 1, 2, 3........etc.

The smallest x = 506629

506629 mod 105 == 4

 Dec 16, 2020

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