# STEP 1 Q1 part 2

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?

(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.

### Year of question paper was

Year of question paper was 1999.

### OK, I get it.

OK, I get it.
What threw me off seems to be the fact that I had never thought of divisibility in this way.
I had been used to tricks like adding digits or checking last digit and whatnot too much.
I'm glad I found this exercise.

### Great!

Great!

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