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I have been reading about the CRT and noticed the following pattern.
Let's say that we are mapping the number $N \pmod {21}$ to the corresponding congruences $\pmod 7$ and $\pmod 3$.
For convenience I tabulate this:

0 1 2 3 4 5 6
0 0 15 9 3 18 12 6
1 7 1 16 10 4 19 13
2 14 8 2 17 11 5 20

This just means that e.g. $10 \equiv 10 \pmod{21}$ and $10 \equiv [3,1]\pmod{[7,3]}$
Now I noticed per row the columns increment by $15 \pmod {21}$ from left to right or by $6 \mod {21}$ from right to left.
I noticed that e.g. for the case of $35 = 7\cdot5$ the columns again increment by $15 \pmod {21}$ from left to right or $20 \pmod {21}$ from right to left.

0 1 2 3 4 5 6
0 0 15 30 10 25 5 20
1 21 1 16 31 11 26 6
2 7 22 2 17 32 12 27
3 28 8 23 3 18 33 13
4 14 29 9 24 4 19 34

I was wondering if that is a real pattern, if there is a specific theorem related to it or if it something we could use to quickly map from $\pmod {21}$ to $\pmod 7$ and $\pmod 3$. Although I don't understand how the increments by $15$ appear

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1 Answer 1

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Yes, as if you go over in a row ,you have to go up by a multiple of $3$ but you need that multiple of $3$ to be $1$ modulo $7$ to factor in the change of column. Similarly, to go down a column, you need it to be a multiple of $7$ that is $1$ modulo $3$ .

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