CONGRUENCES AND MODULAR ARITHMETIC Congruence 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) Ex. Since, , a, b, n ∈ I, n ≠ b 42 ≡ 30 (mod 3) 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: Congruence mod m is an equivalent relation: Equivalence relation is 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 1. 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. 3. Reflexivity: Ex. 30 ≡ 3 (mod 9) 9 ∣ 30 – 3 a ≡ a (mod n) 7 ≡ 7 (mod 1) 1∣ 7– 7 or 14 ≡ 5 (mod 2) 2 ∣ 14 – 5

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 n 1) and a ≡ b (mod n 2) → a ≡ b (mod [n 1, n 2] ), where [n 1, n 2] 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 n 9. ak ≡ bk (mod n) → a ≡ b (mod (k, n) ), where (k, n) is the HCF Ex. 2 ) (4, 2) → 15 ≡ 13 (mod 2 ) 2 15 (4) ≡ 13 (4) (mod 2) → 15 ≡ 13 (mod 60 ≡ 52 (mod 2) → 15 ≡ 13 (mod 1) 2 ∣ 60 – 52 → 1 ∣ 15 – 13

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) 6. Solve the following. 42 ≡ 6 (mod 4) and 42 ≡ 6 (mod 9) 23 (12) ≡ 15 (12) (mod 4) 32 2 ≡ 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