# Is this a fun math fact?

I just came across this and felt compelled to share. You do not need to be a math geek to appreciate it, though it probably helps.

Take a prime number other than 2 or 5, 7 for example. Now look at one over the number, 1/7 in our case. As I hope you recall, this will be a repeating decimal. For prime numbers, the sequence will always be of the form .a1 a2…an a1a2…an…, and so on.

1/7 = .142857142857… If the number of digits in the repeating sequence is even, take the last half and add it to the first half, 142 + 857 = 999, and you will always get all nines.

This is known as Midy’s Theorem. For the mathematically inclined, this Wikipedia article has a nice description.

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## 7 Answers

I remember talking about this in a thread a while back. With 7ths, I like how the repeating sequence (i.e. 1/7=0.142857…) the next fraction is the same permutation, but starting on the next greatest number.

1/7 = 0.142857 (1 is smallest)

2/7 = 0.285714 (2 is next largest)

3/7 = 0.428571 (4 is next largest)

and so on…

I looked it up more, and it works with other primes as well, but sometimes with more than one sequence. For example, 13ths have two: 153846 and 076923. Interestingly, they don’t alternate.

1/13=0.076923…; repetend #1 (r1)

2/13=0.153846…; repetend #2 (r2)

Here’s all 13ths and their sequences:

01/13: r1

02/13: r2

03/13: r1

04/13: r1

05/13: r2

06/13: r2

07/13: r2

08/13: r2

09/13: r1

10/13: r1

11/13: r2

12/13: r1

Interestingly, they don’t alternate, but is it symmetric going up and down!

There are so many neat tricks

Any two digit number multiplied by 11:

90 X 11 = 9_(9+0)_0 = 990

52 X 11 = 5_(5+2)_2 = 572

25X11 = 2_(2+5)_5 = 275

@ARE_you_kidding_me I just figured out that if he number you are multiplying with 11 has more than 2 digits (3, for example) you do something similar. With three, you add the first two, and then the last two. So 436*11= 4_(4+3)_(3+6)_2=4792. Originally, if I was presented with a multiplication problem **n*11**, I’d do **n*10 + n**, but seems easier.

Am I spamming this thread? If so I apologize—I just haven’t talked about numbers a lot recently.

@LostInParadise Very cool! I had not heard of this . Thanks for the introduction.

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