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Percentage of Numbers Divisible by 6, 8

```Date: 03/18/2003 at 20:45:24
From: Mark
Subject: Divisibility by 6, by 8

What percentage of numbers is divisible by 6?

What percentage of numbers is divisible by 8?

I really don't understand the principle of the question.  For
instance, what numbers?
```

```
Date: 03/19/2003 at 08:51:18
From: Doctor Rick
Subject: Re: Divisibility by 6, by 8

Hi, Mark.

The answer does depend on "what numbers," but maybe not as much as
you'd expect.

If it means all real numbers, the answer is zero percent, because
only integers can possibly be divisible by 6, and the ratio of
integers to real numbers is zero. We say that the integers constitute

Countable Sets and Measure Zero
http://mathforum.org/library/drmath/view/51860.html

The connection between this notion and probability (which is related
to your "percentage" question) can be seen here:

Question on Probability of Repeating Digits
http://mathforum.org/library/drmath/view/52212.html

But let's get on to what is really in view in this problem. What
if "numbers" means "integers"? A non-zero percentage of all integers
is divisible by 6, even though the "part" (the number on top in your
percentage calculation) and the "whole" (the number on the bottom)
are both infinite. How can this be?

Consider any finite contiguous set of integers, let's just say from 1
through N. Then there are N numbers in the "whole." How many are there
in the "part"? Does the percentage depend on N? Just slightly, and
that dependence gets smaller as N increases. If you take the limit as
N goes to infinity (so that you're considering all whole numbers),
you'll have a specific value for the percentage.

That's an outline of what's involved. You'll have to decide how much
detail you need to go into, depending on what sort of class this is
for. To me the answer is intuitively obvious, but to prove it
rigorously would take more work.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus
High School Number Theory
High School Probability
High School Sets

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