Repeating Decimals

Awhile back as a class we investigated repeating decimals. After some work and discussion, we came to see that when doing the long division of the fraction, if you arrive at the same remainder twice then you have a repeating decimal. For example consider 1/6:

Each reminder will keep being a multiple of 4, which 6 will continue to go into by the same amount each time. Pretty cool.

Then we all went home with the task of investigating repeating decimals further on our own.

What immediately came to my mind to look into was if there were a way to look at the fraction and tell if it would be repeating decimal without doing the long division or typing it into your calculator. My first thought was just that maybe it dealt with the denominator being prime, since I had been focusing on a 1/3 for some time. But this was quickly, like the second after thought, proven wrong. There's a 1/5 and a 1/2, and 1/6 repeats but 6 is not prime. Then I realized I should probably come up with a couple more examples of repeating decimals, so know what the numbers look like. Here's the list I got: 1/3, 1/6, 1/7, 1/9, 1/11, 1/12, 1/13, 1/14, 1/15, 1/17, 1/18.

I also knew that certain multiples of these would become non-repeating (so I would have to look at fractions in simplified form). I noticed from my list that 1/6, 1/9, 1/12, 1/15, and 1/18 related to a 1/3 since multiples of them could be reduced to a third. They are also was of breaking down a 1/3 into smaller pieces, so it makes sense that they would be repeating decimals as well. I then looked at 1/7 and thought to myself that anything with a denominator a multiple of 7 will be a repeating decimal. I then checked: 1/14, 1/21, 1/28. What do you know, it worked. So it would likewise follow for 11, 13, 17, and 19 (the denominators of other repeating decimals I recorded).

I then noticed something all these numbers had in common, their denominators were composed of more than just 5 and 2 (the prime factors of 10). Those composed of only 5 and 2 or that could be simplified to such were not repeating: 1/5, 1/2, 1/4, 1/8, 1/10, 1/20.

I was curious as to why and might explore that later. I had a thought that since the numbers the denominator is compose of go into 10, when doing the long division you would eventually have a point when that value goes into the reminder (a multiple of 10) completely. But I was content with my findings and had fun processing through it: asking different questions, gathering more examples, analyzing what I had, and then being able to make a conclusion and reach an understanding on what I had set out on,