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Number Theory Fundamentals: Divisibility, Congruences, and Chinese Remainder Theorem - Pro, Lecture notes of Mathematical logic

University of VictoriaMathematical logic
Prof. Gary Macgillivray

The basics of number theory, including divisibility, the euclidean algorithm, congruences, the chinese remainder theorem, and the fundamental theorem of arithmetic. It also discusses prime and composite numbers, fermat's little theorem, and the computation of gcds and lcms.

Typology: Lecture notes

2022/2023

Uploaded on 04/11/2024

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- a divides b: a | b，i.e. b is divisible by a
- a divided by b means a/b
- a: divisor or factor of b; b: divident or multiple of a
- b div a = q (quotient), b mod a = r (remainder)
Property of divisibility:
Let a, b and c be integers with a != 0,
- if a|b and a|c, then a|(b+c)
- if a|b then a|bc
- if a|b and b|c, then a|c
more general: if a|b and a|c, then a|mb+nc
e.g.: a = bq + r, if d | a and d | b, then d | r (since r = a - bq, i.e. d | a - bq)
- m and n co-prime, for any x such that m|x and n|x, we have mn|x
Euclidean division:
Let a, b be integers with a != 0, there exists unique integers q and r such that: b = aq + r (0 <= r < a)
- b div a = q
- b mod a = r
Congruence:
a ≡ b modulo m (positive integer) if and only if:
- m | a-b (also m | b-a)
- a mod m = b mod m
- exists an integer k such that a = b + km (also b = a + k'm)
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Download Number Theory Fundamentals: Divisibility, Congruences, and Chinese Remainder Theorem - Pro and more Lecture notes Mathematical logic in PDF only on Docsity!

  • a divides b: a | b，i.e. b is divisible by a
  • a divided by b means a/b
  • a: divisor or factor of b; b: divident or multiple of a
  • b div a = q (quotient), b mod a = r (remainder) Property of divisibility: Let a, b and c be integers with a != 0,
  • if a|b and a|c, then a|(b+c)
  • if a|b then a|bc
  • if a|b and b|c, then a|c more general: if a|b and a|c, then a|mb+nc e.g.: a = bq + r, if d | a and d | b, then d | r (since r = a - bq, i.e. d | a - bq)
  • m and n co-prime, for any x such that m|x and n|x, we have mn|x Euclidean division: Let a, b be integers with a != 0, there exists unique integers q and r such that: b = aq + r (0 <= r < a)
  • b div a = q
  • b mod a = r Congruence: a ≡ b modulo m (positive integer) if and only if:
  • m | a-b (also m | b-a)
  • a mod m = b mod m
  • exists an integer k such that a = b + km (also b = a + k'm)

If a ≡ b mod m and c ≡ d mod m, then a + c ≡ b + d mod m, and ac ≡ bd mod m More generally:

  • a ≡ b mod m --> d + a ≡ d + b mod m
  • a ≡ b mod m --> ac ≡ bc mod m
  • a ≡ b mod m and c ≡ b mod m, then a ≡ c mod m Note: Division does not always maintain congruences (a + b) mod m = ((a mod m) + (b mod m)) mod m (a * b) mod m = ((a mod m) * (b mod m)) mod m Addition modulo m: a +_m b = (a + b) mod m Multiplication modulo m: a *_m b (a * b) mod m Working modulo m for Z_m = {0, 1, 2, ..., m-1} As a binary relation on Z, congruence modulo m is equivalence relations, i.e. is reflexive, symmetric and transitive. The equivalence class is now called congruence class. Congruence class modulo m of an integer x is the set of all integers congruent to x modulo m: x_bar or [x] = {a ∈ Z | a ≡ x mod m} An integer is congruent modulo m to its remainder on division by m. There are m congruence classes modulo m, each corresponding to the m possible remainders.
  • let m = 5, the set of all equivalence classes moudulo m are {0_bar, 1_bar, 2_bar, 3_bar, 4_bar}, this set and Z_5 = {0, 1, 2, 3, 4} are isomorphic.
    • more general, let mZ = {..., - 2m, - m, 0, m, 2m,...}, the set of congruence class modulo m are: 0_bar = mZ; 1_bar = mZ + 1, ..., (m-1)_bar = mZ + (m-1)

350 = 25^2

gcd(1470,350) = 2^min(1,1) * 3^min(1,0) * 5^min(1,2) * 7^min(2,1) = 70 LCM (least common multiple): The least common multiple of two positive integers a and b is the smallest positive integer that is divisible by both a and b.

  • compute LCMs from primes: 1470 = 2357^ 350 = 25^2* lcm(1470,350) = 2^max(1,1) * 3^max(1,0) * 5^max(1,2) * 7^max(2,1) = 7350 For any two positive integers a and b: a * b = gcd(a,b) * lcm(a,b) Let a and b be integers with a = bq + r (0 <= r < b) by Euclidean division: gcd(a,b) = gcd(b,r) = gcd(b,a−qb) compute GCDs using Euclidean algorithm: repeated applications of Euclidean division Bezout's theorem: for any positive integers a and b, there exists integers s and t such that gcd(a,b) = sa + tb (the fomulas: Bezout identity; a and b: Bezout coefficients)
  • Let a, b, c be positive integers such that a and b are relatively prime. If a|bc then a|c
  • Let p be a prime integer and a1, a2, ..., a_n be integers. If p | a1a2...a_n, then p | a_i for at least one i.
  • Let m be a positive integer and a, b, c be integers. If gcd(c,m) = 1 and ac ≡ bc mod m, then a ≡ b mod m (over the real): a multiplicative inverse of a number is another number such that their product equals 1 (working modulo a number): given number a and modulus m, the modular multiplicative inverse of a is another number b such that ab ≡ 1 mod m

If a and m are relatively prime integers with m > 1, then there exists a unique modular inverse x of a modulo m satisfying 0 < x < m Note: when a and m are not relatively prime (i.e. gcd(a,m) != 1, then a doesn't have modular inverse, and a is called a zero-divisor for arithmetic modulo m) Chinese remainder theorem (CRT): let m, n be two co-prime integers greater than 1, then: ⌈x ≡ a mod m ⌊x ≡ b mod n has a unique solution modulo m * n: sm + tn = 1, then x = bsm + atn (≡ a mod m ≡ b mod n)