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Subject: "Chinese Remainder Theorem"     Previous Topic | Next Topic
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Conferences The CTK Exchange College math Topic #74
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gh@over-yonder.net (Guest)
Mar-12-01, 01:42 PM (EST)
"Chinese Remainder Theorem"
   I read the proof, but I am lost in the solution to problem one.
Problem 1, on <https://www.cut-the-knot.com/blue/chinese.shtml>


p1: x = 2 (mod 3)
p2: x = 3 (mod 5)
p3: x = 2 (mod 7)

From p1, x = 3t 2, for some integer t. Substituting this into p2 gives 3t = 1.

I see that
x = 3t 2 = 5s 3, which gives,
3t - 5s = 1, but how does that yield
3t = 1 ?

Thank you for helping me to understand this.
(Note that I am only in high school, which may contribute to the lack of knowledge which has prevented me from understanding this; and for that I apologize.)


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Charter Member
672 posts
Mar-13-01, 00:12 AM (EST)
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1. "RE: Chinese Remainder Theorem"
In response to message #0
>p1: x = 2 (mod 3)
>p2: x = 3 (mod 5)
>p3: x = 2 (mod 7)
>From p1, x = 3t
> 2, for some integer
>t. Substituting this into p2
>gives 3t = 1.
>I see that
>x = 3t + 2 =
>5s + 3, which
>3t - 5s = 1, but
>how does that yield
>3t = 1 ?

If x = 3t + 2 and x = 3 (mod 5) then
3t + 2 = 3 (mod 5), or
3t = 1 (mod 5).

This is what substitution is about. Or you can refer to the transitive law:

if A = B and B = C, then A = C.

It matters not whether the equality here is the "real" equality or equality modulo some number.

You may also continue in your way. Except that you should note that what is asserted is that

3t = 1 (mod 5)

(Now that I had a look at the page, it really says "3t = 1", which is of course not true. I am going to fix this. You have my appreciation for detecting the mistype.)

3t - 5s = 1 is true in general, but modulo 5 we get

3t = 1 (mod 5)

since obviously 5s = 0 (mod 5).

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