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)

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 equivalence 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:

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 !!

Exercise • 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)

Exercise • 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 

CONGRUENCES AND MODULAR ARITHMETIC