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Peter Burkhardt 1

Number Theory

Lecture 1

Divisibility and Modular Arithmetic (Congruences)

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Basic Definitions and Notations (1)

N = {1,2,3,…}

denotes the set of natural numbers

Z = {…,-3,-2,-1,0,1,2,3,…} denotes the set of integers

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Basic Definitions and Notations (2)

Divisibility (1)Let a, b Z, a not equal to zero.We say a divides b if there existsan integer k such that

akb akb

akb

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Basic Definitions and Notations (3)

Divisibility (2)In this case we write

a|bSometimes we say that: b is divisible by a, or a is a factor of b, or b is a multiple of a

akb akb

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Basic Definitions and Notations (4)

Prime and Composite NumbersA natural number p > 1 is called a prime number, or, simply, prime, if it is divisible only by itself and by 1.

P = {2,3,5,7,…} denotes the set of prime numbers.Otherwise the number is called composite.

akb

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Properties of Divisibility

a|b ”a|bc for each integer c a|b and b|c ”a|c a|b and a|c ”a|(bx + cy) for any x, y Z a|b and a, b not equal to zero ”|a| |b|

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Division with Remainder

Let m ,a Z, m > 1.

Then, there exist uniquely determined

numbers q and r such that

a = qm + r

with

0 r < m

Obviously, m|a if and only if r = 0.

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Congruences

Let a, b Z, m N. We say

a is congruent to b modulo m

if

m|(a-b)

and we write

)( shorter, or, )(mod mbamba

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Congruence and Division with Remainder

)(mod mba

)(mod mba

Dividing a and b by m yields the same remainder.

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Basic Properties of Congruences

)(mod)(mod),(mod :tyTransitivi

)(mod)(mod :Symmetry

)(mod :yReflexivit

: ,,,

mcamcbmba

mabmba

maa

NmZcba

That is, congruence is an equivalence relation.

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Modular Arithmetic

Nkmba

mdbca

mdbca

mdcmba

NmZdcba

kk

)(mod :RulePower

)(mod :RuleProduct

)(mod :RuleAddition

then),(mod),(mod If

: ,,,,

0for )(mod :RuleFactor

)(mod :RulePower

)(mod :RuleProduct

)(mod :RuleAddition

then),(mod),(mod If

: ,,,,

ccmcbca

Nkmba

mdbca

mdbca

mdcmba

NmZdcba

kk

”Demonstration

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Little Fermat’s Theorem

)(mod

)(mod11

paa

pap

p

)(mod11 pa p

Let a Z, and p prime. If p does not divide a, then

For all a Z we have

)(mod paa p

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Have you understood?

How can you write the following statements

using congruences? (a, b, r Z, mN)

1. m|a

2. r is the remainder of a divided by m

Using congruences, give a sufficient condition

for

m|a if and only if m|b

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Practice (Handouts)