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# What does modulo mean?

Posted on 2011-09-08
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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
• 5
• 3
• 2
• +1

Author Comment

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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LVL 84

Expert Comment

yes
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LVL 84

Expert Comment

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

Can you explain the -9 part in your post? I didn't understand that part.
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LVL 84

Expert Comment

10 divides 1 - -9
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LVL 84

Accepted Solution

ozo earned 500 total points
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

thanks
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LVL 31

Expert Comment

@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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LVL 37

Expert Comment

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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LVL 37

Expert Comment

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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LVL 31

Expert Comment

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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LVL 84

Expert Comment

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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