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What does modulo mean?

Posted on 2011-09-08
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Last Modified: 2012-05-12
In my Data Structures textbook it says:

"We say that A is congruent to B modulo N, written A = B (mod N), if N divides A - B. Intuitively this means that the remainder is the same when either A or B is divided by N. Thus, 81 = 61 = 1 (mod 10)"

Does modulo mean that division using the number N yields the same remainder on both A and B?
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Question by:shampouya
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12 Comments
 

Author Comment

by:shampouya
ID: 36507065
The equals signs above should have three lines, not two. I couldn't get the congruency symbol to work in this browser.
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Expert Comment

by:ozo
ID: 36507072
yes
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Expert Comment

by:ozo
ID: 36507085
provided that the sign of the remainder does not depend on the sign of A or B, so that
-9 = 81 = 61 = 1 (mod 10)
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Author Comment

by:shampouya
ID: 36507101
Can you explain the -9 part in your post? I didn't understand that part.
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Expert Comment

by:ozo
ID: 36507140
10 divides 1 - -9
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Accepted Solution

by:
ozo earned 500 total points
ID: 36507147
But some division operators are defined so that the remainder is the same sign as the dividend.
the "remainder is the same" intuition works better with division operators that are defined so that the remainder is the same sign as the divisor
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Author Closing Comment

by:shampouya
ID: 36507166
thanks
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Expert Comment

by:Paul Sauvé
ID: 36507239


@ozo: if I understand correctly, provided that the sign of the remainder does not depend on the sign of A or B -9 mod 10 = 9, but -11 mod 10 = 1?
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Expert Comment

by:TommySzalapski
ID: 36507467
No paul, it's a straight pattern
5 mod 3 = 2
4 mod 3 = 1
3 mod 3 = 0
2 mod 3 = 2
1 mod 3 = 1
0 mod 3 = 0
-1 mod 3 = 2
-2 mod 3 = 1
-3 mod 3 = 0
-4 mod 3 = 2

So the negatives are sort of backwards to the positives.
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Expert Comment

by:TommySzalapski
ID: 36507480
In some places it goes like this:
5 mod 3 = 2
4 mod 3 = 1
3 mod 3 = 0
2 mod 3 = 2
1 mod 3 = 1
0 mod 3 = 0
-1 mod 3 = -1
-2 mod 3 = -2
-3 mod 3 = 0
-4 mod 3 = -1

But if you add the divisor to the mod, it gives the same answer as before.

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Expert Comment

by:Paul Sauvé
ID: 36507509
I actually learned that at university in my IT courses back in the late 70's! My RAM is gettint a little screwed up! Too mush stuff in there!
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Expert Comment

by:ozo
ID: 36508246
5 mod 3 = 2
4 mod 3 = 1
3 mod 3 = 0
2 mod 3 = 2
1 mod 3 = 1
0 mod 3 = 0
-1 mod 3 = 2
-2 mod 3 = 1
-3 mod 3 = 0
-4 mod 3 = 2
So the negatives are sort of backwards to the positives.

I'd say that the negatives follow exactly the same pattern as the positives
Whereas
In some places it goes like this:
5 mod 3 = 2
4 mod 3 = 1
3 mod 3 = 0
2 mod 3 = 2
1 mod 3 = 1
0 mod 3 = 0
-1 mod 3 = -1
-2 mod 3 = -2
-3 mod 3 = 0
-4 mod 3 = -1
seems to be "backwards"

To me it makes no sense to have 5 different mod 3 classes

a mod b can be defined as
a - b*floor(a/b)
which works consistently for any sign of a or b, and even for non-integer a or b

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