# Are these incremements a valid/useful pattern when mapping between modulo (crt)

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

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 $$3$$ .