My First Maths Question - Modulus Arithmetic

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My First Maths Question - Modulus Arithmetic

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Hi,
This is my 1932'nd post and this is the first 'real' maths question that i have asked
Since, Reinout-g, I like Serena and Rom have been answering so many questions lately i have had time to go learn some things and get some questions of my own
I'm am looking at solving this:

5/86 = k Mod8 (I made it up, that's probably why it doesn't work) (k is in the set 0,1,2,3,4,5,6,7 (since it is mod 8))

This is my working
5=86k mod 8
86k=5 mod 8
86k=8a+5 for some integer value a
(86k-5)/8 = a
86k/8 -5/8 = a
10k + 6k/8 - 5/8 = a
10k + (6k-5)/8 = a
Since a is an integer 6k-5 must be a multiple of 8
6k-5 = 8N for some integer N
therefore
6k-5 must equal one of the following 0,8,16,24,..........
so 6k must equal one of the following 5,13,21,29,37,45,53,61,69,77
but none of these numbers are exactly divisible by 6.
Does this mean that there is no solution, or have i done something wrong?
Thankyou

Melody Jan 31, 2014

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I like Serena · Answer 1 · 2014-01-31T08:24:13

Hey Melody!

Let me see if I can shed some light here.

Melody:
5/86 = k Mod8 (I made it up, that's probably why it doesn't work) (k is in the set 0,1,2,3,4,5,6,7 (since it is mod 8))

This is my working
5=86k mod 8

(snip)

so 6k must equal one of the following 5,13,21,29,37,45,53,61,69,77
but none of these numbers are exactly divisible by 6.
Does this mean that there is no solution, or have i done something wrong?

5 is odd, while 86k is even.
If we add a multiple of 8 to 86k, it is still even!
So indeed there is no solution in the whole numbers.

Btw, note that 86k = 88k -2k = -2k mod 8.
So 5=86k mod 8 simplifies to 5=-2k mod 8.

Now let's see what Reinout-g's calculator says:

[input]5 = 86k mod 8[/input]
[input]6 = 86k mod 8[/input]
Hah!

Melody · Answer 2 · 2014-01-31T09:10:31

Thanks I like Serena,
I never did any modulus arithmetic when I was younger. It is all completely new to me.
I really don't think I understand the concepts much at all.
I was told elsewhere that modulus arithemetic only used whole numbers and now you are introducing fractions.
I'll think about what you have told me, but I really don't get it.
Don't be too concerned, it is not important, I am only trying to fluff my knowledge out at the edges. Like the tassles on a mat.
If it is too difficult I lose interest very easily.
Stay happy,
Melody.

I like Serena · Answer 3 · 2014-01-31T09:21:16

Melody:
Thanks I like Serena,
I never did any modulus arithmetic when I was younger. It is all completely new to me.
I really don't think I understand the concepts much at all.
I was told elsewhere that modulus arithemetic only used whole numbers and now you are introducing fractions.
I'll think about what you have told me, but I really don't get it.
Don't be too concerned, it is not important, I am only trying to fluff my knowledge out at the edges. Like the tassles on a mat.
If it is too difficult I lose interest very easily.
Stay happy,
Melody.

You're quite right. Modulus arithmetic does not involve fractions.
I just had to smile when this forum's calculator came up with a fraction.
And also when it failed to simplify the second equation.
Wolfram Alpha does a much better job.

Check out these links:
http://m.wolframalpha.com/input/?i=5%3D86k+mod+8
http://m.wolframalpha.com/input/?i=6%3D86k+mod+8

When you wrote 5/86, that should be interpreted as 5 x 86 ^-1.
Here 86 ^-1 should be interpreted as the number when multiplied by 86 is 1 modulus 8.
It so happens that this number does not exist (that is because 86 and 8 have a factor in common).

Melody · Answer 4 · 2014-02-01T06:53:33

Thanks i like Serena
I'll do that.

My First Maths Question - Modulus Arithmetic

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