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

Question

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

Answer 1

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 $3$ .