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CONGRUENCES AND MODULAR ARITHMETIC

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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)

Since, 3 ∣ 42 – 30

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

Ex. 3 ≡ 4 (mod 5)

- 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)

- 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)

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:

- 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)

- 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

- 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

- 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

- 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

EXERCISE !!

- 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)

- 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)

THANK YOU

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