Opsimath

2021/09/16阅读：49主题：默认主题

# STEP 4

The question today comes from 1999 Paper I. The original question is shown below.

(i) How many integers greater than or equal to zero and less than 1000 are not divisible by 2 or 5? What is the average value of these integers?

(ii) 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?

There are actually two methods for doing this. I will first give a regular solution (by creating "blocks of 10 numbers") then provide you with an interesting method. (Even more interesting is the fact that these two methods are, to a certain degree, equivalent.)

*Solution 1*. (i) For any integers between
and
where
, there are exactly 4 numbers not divisible by 2 or 5 (namely,
,
,
, and
). The number of integers is therefore
. The integers can be added in pairs:
since there are 200 such pairs, giving an average of 500.

(ii) For any integers between and where , there are exactly 12 numbers not divisible by 3 or 7. The number of integers is therefore . The integers can be added in pairs: since there are 1323 such pairs, giving an average of .

*Solution 2*. (i) Since both 2 and 5 are prime numbers, removing multiples of one number does not affect the proportion of multiples of the others. So the answer is simply
. The average is calculated by the same method above.

(ii) Since both 3 and 7 are prime numbers, removing multiples of one number does not affect the proportion of multiples of the others. So the answer is simply . The average is calculated by the same method above.

*Extension*. Why 10? Why 21? Notice that
and
. What is the significance of this? Can you generalize the arguments to the case "How many integers greater than or equal to zero and less than
are not divisible by
or
"?

There is a (similar?) puzzle: How many houses a postman delivers to in a street of houses numbered from 1 to 1000, given that he or she refuses to deliver to houses with the digit 9 in the number?

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