The question says:
How many integers greater than or equal to zero and less than 9261 are not divisible by 3 or 7? What is the average value of these integers?

The answer says:
(ii) Either of the above methods will work. In the first method you consider integers in blocks of 21 (essentially arithmetic to base 21): there are 12 integers in each such block that are not divisible by 3 or 7 (namely 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20) so the total number is 9261 x 12=21 = 5292. The average is 9261=2 as can be seen using the pairing argument (1+9260)+(2+9259)+... or the symmetry argument.

I'm having difficulties understanding the 21 blocks argument. I understand 3x7=21, but how does that help in answering this question? How could I show that indeed every block of 21 integers contains exactly 12 integers not divisible by 3 or 7?

Would this technique work with any pair of integers? e.g. 87 and 13 to produce a block of 1131?

Thanks.