A student wants to know how many integers between 1 and 1000 are a multiple of
3 or a multiple of 5. She wonders if it is correct to find the number of those
integers that are multiples of 3 and add the number of those that are multiples
of 5. How do you respond?
No because if you added those you would be counting all multiples
of 15 twice, because they are both multiples of 3 as well as multiples
of 5.
Here is what you have to do:
To find the number of multiples of positive integer k between 1 and
positive integer N of any positive integer:
Divide N by k and take only the integer part.
So to find the number of multiples of 3 between 1 and 1000,
we divide 1000 by 3, getting 333.3333333, and take the
integer part. So there are 333 multiples of 3 between 1 and
1000.
To find the number of multiples of 5 between 1 and 1000,
we divide 1000 by 5, getting 200, and taking the integer part,
we also have 200. So there are 200 multiples of 5 between 1
and 1000.
However if we add these 333+200, this number, 533, counts every
integer twice which is both a multiple of 3 as well as a multiple
of 5.
Every integer which is a multiple of 3 as well as a multiple of 5
is also a multiple of 15. So to figure how many integers we have
counted twince in 533,
we find the number of multiples of 15 between 1 and 1000,
we divide 1000 by 15, getting 66.66666667, and taking the integer part,
we get 66. So there are 66 multiples of 15 between 1 and 1000.
Since 533 counts these 66 twice, we merely subtract 533-66, getting
467 which counts them only once.
Actually we used the formula
N(A or B) = N(A) + N(B) - N(A and B)
= 333 + 200 - 66
= 467
where A is the number of multiples of 3 and B is the number of
multiples of 5.
Edwin
