
Originally Posted by
DivideByZero
When counting by 5s what ratio of numbers are divisible by 2 and 3?
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80...
Be careful with your language. The numbers in the list above divisible by 2
and 3 are the multiples of 30. The numbers you put in bold are those divisible by 2
or 3.
Let a and b be integers. A number n is divisible by a and b if and only if it is divisible by their least common multiple (lcm), which I denote [a,b]. So if I list out the multiples of a (i.e., if I count by a) and ask which ones are divisible by b, I obtain the multiples of [a,b]. Now it shouldn't be too hard to see that the ratio of multiples of [a,b] to multiples of a is a/[a,b].
We can extend this further. For example, for three integers a, b, and c, let's list out the multiples of a. Which ones are also divisible by b and c? Precisely the ones divisible by [b,c]. So then the ones divisible by a, b, and c are the multiples of [a,[b,c]] = [a,b,c], the least common multiple of a, b, and c. Then the ratio of numbers divisible by a, b, and c to those divisible by a is a/[a,b,c].
We can also answer the question: given the multiples of a, which ones are also divisible by b or c (or both)? Well, we know that a/[a,b] of them are divisible by a and b and a/[a,c] of them are divisible by a and c. If I add these numbers together, I count some numbers too many times. To be precise:
-if m is divisible by a and b but not c, it's only counted in a/[a,b]
-if m is divisible by a and c but not b, it's only counted in a/[a,c]
-if m is divisible by a, b, and c, it's counted in both a/[a,b] and a/[a,c]--i.e., it's counted twice
So to get the right answer, we have to subtract the numbers divisible by a, b, and c to account for them being counted twice. But we just figured this out. So our answer is:
a/[a,b]+a/[a,c]-a/[a,b,c]
I just realized--this means your 1/3 is wrong. If we count just numbers divisible by 2 and 3 out of those divisible by 5, we should get 5/[2,3,5] = 5/30 = 1/6. If we count numbers divisible by 2 or 3 out of those divisible by 5, we get:
5/[2,5]+5/[3,5]-5/[2,3,5] = 5/10+5/15-5/30 = 1/2+1/3-1/6 = 2/3
and this matches up with your list.