Chinese Remainder Theorem

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• May 10th 2010, 12:45 PM
GraceMahon
Chinese Remainder Theorem
Hi (Talking)

Hi,

Could anyone help with this proof? I need to use the chinese remainder theorem, but have no idea where to start? Does anyone have any ideas?

To check if f(x, y) = 0 (mod m) has integer solutions where
m > 1 and m in Z. It suffces to check it for prime powers, where m = p^k where k is bigger than or greater than 1
and where p is a prime.

Any ideas or hints would be great!

Thanks!
• May 10th 2010, 01:24 PM
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Quote:

Originally Posted by GraceMahon
Hi (Talking)

Hi,

Could anyone help with this proof? I need to use the chinese remainder theorem, but have no idea where to start? Does anyone have any ideas?

To check if f(x, y) = 0 (mod m) has integer solutions where
m > 1 and m in Z. It suffces to check it for prime powers, where m = p^k where k is bigger than or greater than 1
and where p is a prime.

Any ideas or hints would be great!

Thanks!

Look at the statement of the theorem, and use the fact that $p_i^{k_i}$ are pairwise coprime. If you don't see it, try looking at some smaller numbers (for example, take any multiple of 2^3 * 3^2 * 5 * 7 and then look at the congruence mod (2^3), mod (3^2), etc.). You should see a very definite pattern.