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CONGRUENCES AND MODULAR ARITHMETIC. Congruence and Modular Arithmetic Definition: a is congruent to b mod n means that n∣a -b, (a-b) is divisible by n. Notation: a ≡ b (mod n) , a, b, n ∈ I, n ≠ b Ex . 42 ≡ 30 (mod 3)

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CONGRUENCES AND MODULAR ARITHMETIC

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Congruences and modular arithmetic

CONGRUENCES

AND

MODULAR ARITHMETIC


Congruences and modular arithmetic

Congruence and Modular Arithmetic

Definition:a is congruent to b mod n means that n∣a-b, (a-b) is divisible by n.

Notation: a ≡ b (mod n), a, b, n ∈ I, n ≠ b

Ex. 42 ≡ 30 (mod 3)

Since, 3 ∣ 42 – 30

a ≡ b (mod n), it means that n ∣ a – b

Ex. 3 ≡ 4 (mod 5)


Congruences and modular arithmetic

  • Congruence and Modular Arithmetic

  • If two numbers a and b have the property that their difference a-b is divisible by a number n (ex. (a-b) ∣ n is an integer), then a and b are said to be "congruent modulo n." The number n is called the modulus, and the statement "a is congruent to b (modulo n)" is written mathematically as

  • a ≡ b (mod n)


Congruences and modular arithmetic

  • Congruence and Modular Arithmetic

  • If a – b is not divisible by n, then it is said that "a is not congruent to b (modulo n)," which is written as

  • a ≡ b (mod n)


Congruences and modular arithmetic

Proposition

Proposition: Congruence mod m is an equivalent relation:

Equivalence relationis a reflexive(every element is in the relation to itself), symmetric(element a has the same relation to element b that b has to a), and transitive (a is in a given relation to b and b is in the same relation to c, then a is also in that relation to c) relationship between elements of a set.

Proposition: Any relation is called an equivalence relation if it satisfied the following properties:


Congruences and modular arithmetic

  • Proposition

  • Reflexivity (every element is in the relation to itself)

  • a ≡ a (mod n) for all a

  • Ex. 3 ≡ 3 (mod 5)

  • 2. Symmetry(element A has the same relation to element B that B has to A), If a ≡ b (mod n), then b ≡ a (mod n)

  • Ex. 10 ≡ 7 (mod 3), then 7 ≡ 10 (mod 3)

  • 3.Transitivity

  • If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n)

  • Ex. 20 ≡ 4 (mod 8), then 4 ≡ 12 (mod 8), then 20 ≡ 12 (mod 8)


Congruences and modular arithmetic

  • Properties

  • 1. Equivalence:a ≡ b (mod 0) → a ≡ b (which can be regarded as a definition)

  • Ex. 18 ≡ 6 (mod 0)

  • 0 ∣ 18 – 6

  • 2. Determination:either a ≡ b (mod n) or a ≡ b (mod n)

  • Ex. 30 ≡ 3 (mod 9) or 14 ≡ 5 (mod 2)

  • 9 ∣ 30 – 3 2 ∣ 14 – 5

  • 3. Reflexivity:a ≡ a (mod n)

  • Ex. 7 ≡ 7 (mod 1)

  • 1 ∣ 7 – 7


Congruences and modular arithmetic

  • Properties

  • 4. Symmetry:a ≡ b (mod n), then b ≡ a (mod n)

  • Ex. 20 ≡ 2 (mod 6), then 2 ≡ 20 (mod 6)

  • 6 ∣ 20 – 2 , then 6 ∣ 2 – 20

  • 5. Transitivity:

  • a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n)

  • Ex. 16 ≡ 4 (mod 2) and 4 ≡ 8 (mod 2), then 16 ≡ 8 (mod 2)

  • 2 ∣ 16 – 4 and 2 ∣ 8 – 4 , then 2 ∣ 16 – 8

  • 6. a ≡ b (mod n) → (k)a ≡ (k)b (mod n)

  • Ex. 25 ≡ 5 (mod 10) → (2)25≡ (2)5 (mod 10)

  • 10 ∣ 25 – 5 → 10 ∣ 50 – 10


Congruences and modular arithmetic

  • Properties

  • 7. a ≡ b (mod n) → am≡bm (mod n), n ≥ 1

  • Ex. 42 ≡ 12 (mod 3)

  • 3 ∣ 16 – 1

  • 8. a ≡ b (mod n1) and a ≡ b (mod n2) → a ≡ b (mod [n1, n2] ), where [n1, n2] is the LCM

  • Ex. 15 ≡ 3 (mod 4) and 15 ≡ 3 (mod 6)

  • →15 ≡ 3 (mod [4, 6] )

  • →15 ≡ 3 (mod 12)

  • → 12 ∣ 15 – 3


Congruences and modular arithmetic

  • Properties

  • 9. ak ≡ bk (mod n) → a ≡ b (mod ),

  • where (k, n) is the HCF

  • Ex. 15 (4) ≡ 13 (4) (mod 2) → 15 ≡ 13 (mod )

  • 60 ≡ 52 (mod 2) →15 ≡ 13 (mod )

  • →15 ≡ 13 (mod 1)

  • 2 ∣ 60 – 52 → 1 ∣ 15 – 13

n

(k,n)

2

(4,2)

2

2


Congruences and modular arithmetic

EXERCISE !!


Congruences and modular arithmetic

  • Excersise

  • 1. Give an example for transitivity property “a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n)”.

  • 2. Find 3 numbers of “a”

  • a ≡ 10 (mod 3), then 10 ≡ a (mod 3)

  • 3. Find x

  • 24 ≡ 8 (mod x), then 8 ≡ 18 (mod x),

  • then 24 ≡ 18 (mod x)

  • 4. Find 3 numbers of “b”.

  • 38 ≡ b (mod 3) → (2)38≡ (2)b (mod 3)


Congruences and modular arithmetic

  • Excersise

  • 5. True or False

  • 283 ≡ 53 (mod 3)

  • 383 ≡ 53 (mod 3)

  • 255 ≡ 6 (mod 7)

  • Solve the following.

  • 42 ≡ 6 (mod 4) and 42 ≡ 6 (mod 9)

  • 23 (12) ≡ 15 (12) (mod 4)

  • 322 ≡ 83 (mod 6)


Congruences and modular arithmetic

THANK YOU


Congruences and modular arithmetic

Submitted by:

EP 4/1 Group 3

Chosita K. “2”

Hsinju C. “3”

Nipawan P. “5”

Ob-Orm U. “11”

Submitted to:

Mr. Wendel Glenn Jumalon


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