What does modulo mean?

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?
shampouyaAsked:
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ozoConnect With a Mentor Commented:
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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shampouyaAuthor Commented:
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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ozoCommented:
yes
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ozoCommented:
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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shampouyaAuthor Commented:
Can you explain the -9 part in your post? I didn't understand that part.
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ozoCommented:
10 divides 1 - -9
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shampouyaAuthor Commented:
thanks
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Paul SauvéRetiredCommented:


@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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TommySzalapskiCommented:
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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TommySzalapskiCommented:
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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Paul SauvéRetiredCommented:
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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ozoCommented:
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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