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# Interesting property

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I was thinking about this interesting property of positive integers, where if a, b, are positive integers, and a>b, then the remainder when a and b are divided by a-b seems to always the same. Is this true for all positive integers? I can't seem to find a simple intuitive proof for this. I'm also curious if this is an iff relationship, if two positive integers a, b have the same remainder when divided by x, is x always a-b? thanks!

Jun 15, 2019

#1
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Given that  a  and  b  are positive integers where  a > b , I do think it is always true that

a mod (a - b)   =   b mod (a - b)

I don't know the best way to explain it, but here is how I convinced myself. In the diagram,

w  is the remainder when  a  is "filled up" with as many blocks of width  a-b  as possible.

x  is the remainder when  b  is "filled up" with as many blocks of width  a-b  as possible.

In other words,

w  =  a mod (a - b)

x  =  b mod (a - b)

And we can see that...

w + (a - b)  =  x + (a - b)

w  =  x

Therefore,

a mod (a - b)  =  b mod (a - b) But I do not think it is always true that if     a mod  x  =  b mod x     then     x  =  a - b

For example,     10 mod 3  =  4 mod 3     but     3  ≠  10 - 4

Jun 15, 2019
edited by hectictar  Jun 15, 2019
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Thanks Hectictar,

I have been playing around with modular arithmetic lately,

It will take me a while to get my head around this.

I am not asking for more explanation, (not yet anyway)  it is just a compliment. Melody  Jun 16, 2019
#6
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Haha, I'm definitely a beginner at this, but thanks!

Once you see what I did you will understand it easily I'm sure. And I'm still kind of curious about a proof for response #2

hectictar  Jun 16, 2019
edited by hectictar  Jun 17, 2019
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I see. Thanks for all the help! I figured it had something to do with modular arithmetic, but I myself barely know the basics.

Guest Jun 17, 2019
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This pretty much sums up my knowledge of it:  http://mathforum.org/library/drmath/view/55771.html hectictar  Jun 17, 2019
#2
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two positife integers a and b such that a>b have the same remainder when divided by x if and only if x is a factor of a-b

Jun 15, 2019
#3
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I agree with hectictar that your first observation is TRUE, but the second is FALSE. I tested 1,000,000 numbers on the computer and it holds for both observations.

Jun 15, 2019
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Wait.... 1,000,000??????? How? did you test it..... I would like to know the program...

NoobGuest  Jun 16, 2019