We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
+1
110
8
avatar

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
avatar+8720 
+3

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
 #5
avatar+103678 
+1

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.   wink

Melody  Jun 16, 2019
 #6
avatar+8720 
+2

Haha, I'm definitely a beginner at this, but thanks!

 

Once you see what I did you will understand it easily I'm sure. laugh

 

And I'm still kind of curious about a proof for response #2

hectictar  Jun 16, 2019
edited by hectictar  Jun 17, 2019
 #7
avatar
+1

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
 #8
avatar+8720 
+2

This pretty much sums up my knowledge of it:  http://mathforum.org/library/drmath/view/55771.html   smiley

hectictar  Jun 17, 2019
 #2
avatar
+1

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
avatar
+1

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
 #4
avatar+208 
0

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

NoobGuest  Jun 16, 2019

53 Online Users

avatar