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Congruence Modulo

You may see an expression like:

AB(mod C)

This says that Ais congruentto Bmodulo C.

We will discuss the meaning of congruence moduloby

performing a thought experiment with the regular modulo

operator.

Let's imagine we were calculating mod 5 for all of the integers:

uppose we labelled 5 slices!" #" $" %" &. Then" for each of the

integers" we put it into a slicethat matched the alue of the

integer mod 5.

Think of these slices as buckets" which hold a set of numbers. (or

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example" $) would go in the slice labelled 1"

because26 mod 5=1.

*boe is a figure that shows some integers that we would find in

each of the slices.

+t would be useful to hae a way of expressing that numbers

belonged in the same slice. ,-otice $) is in the same slice as

#" )" ##" #)" $# in aboe example.

* common way of expressing that two alues are in the same

slice" is to say they are in the same equivalence class.

The way we express this mathematically for mod /

is: AB(mod C)The aboe expression is pronounced Ais congruent

toBmodulo C.

0xamining the expression closer:

#. is the symbol for congruence" which means the

alues Aand Bare in the same equivalence class.

$. (mod C)tells us what operationwe applied to Aand B.%. when we hae both of these" we call congruence

modulo C.

e.g. 2611 (mod 5)

26 mod 5=1so it is in the e1uialence class for #"

11 mod 5=1so it is in the e1uialence class for #"