Outline
Basic properties of congruence
Modular addition and multiplication
Introduction to Number Theory II
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Basic properties of congruence
Modular addition and multiplication
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Definition
1
Let n > 0 be an integer. We say that integers a and b are
congruent modulo n if n divides their difference.
We write a ≡ b mod n. The number n is called the modulus
and b is called the remainder.
1
Carl F. Gauss, Disquisitiones Arithmeticae, 1801.
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Definition
1
Let n > 0 be an integer. We say that integers a and b are
congruent modulo n if n divides their difference.
We write a ≡ b mod n. The number n is called the modulus
and b is called the remainder.
The remainder b is not unique. Here is an example:
I
12 ≡ 3 mod 9 ; 3 is a valid remainder since 9 divides 12 − 3
I
12 ≡ 21 mod 9 ; 21 is a valid remainder since 9 divides
12 − 21
I
12 ≡ −6 mod 9 ; −6 is a valid remainder since 9 divides
12 − (−6)
1
Carl F. Gauss, Disquisitiones Arithmeticae, 1801.
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Theorem
For arbitrary integers a and b, a ≡ b (mod n) if and only if
a mod n = b mod n.
Proof: Exercise.
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Lemma
The following hold for n ≥ 1:
I
a ≡ a (mod n)
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Lemma
The following hold for n ≥ 1:
I
a ≡ a (mod n)
I
a ≡ b (mod n) implies b ≡ a (mod n)
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Lemma
The following hold for n ≥ 1:
I
a ≡ a (mod n)
I
a ≡ b (mod n) implies b ≡ a (mod n)
I
a ≡ b (mod n) and b ≡ c (mod n) implies
a ≡ c (mod n)
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Lemma
The following hold for n ≥ 1:
I
a ≡ a (mod n)
I
a ≡ b (mod n) implies b ≡ a (mod n)
I
a ≡ b (mod n) and b ≡ c (mod n) implies
a ≡ c (mod n)
I
a ≡ b (mod n) implies ak ≡ b k (mod n) for any positive
integer k.
Proof: Exercise.
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Definition
The set Zn = {0, 1, 2, 3, ..., n − 1} has two operations ”+” and ”·”
for all a, b ∈ Zn such that:
I
a + b ≡ c mod n (c ∈ Zn )
I
a · b ≡ d mod n (d ∈ Zn )
Example
17 + 20 mod 22 = 15 and 4 · 8 mod 22 = 10.
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Lemma
The following hold for n ≥ 1:
I
a ≡ b (mod n) implies a + c ≡ b + c (mod n)
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Lemma
The following hold for n ≥ 1:
I
a ≡ b (mod n) implies a + c ≡ b + c (mod n)
I
a ≡ b (mod n) implies ac ≡ bc (mod n)
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Lemma
The following hold for n ≥ 1:
I
a ≡ b (mod n) implies a + c ≡ b + c (mod n)
I
a ≡ b (mod n) implies ac ≡ bc (mod n)
I
a ≡ b (mod n) and c ≡ d (mod n) imply
a + c ≡ b + d (mod n)
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Lemma
The following hold for n ≥ 1:
I
a ≡ b (mod n) implies a + c ≡ b + c (mod n)
I
a ≡ b (mod n) implies ac ≡ bc (mod n)
I
a ≡ b (mod n) and c ≡ d (mod n) imply
a + c ≡ b + d (mod n)
I
a ≡ b (mod n) and c ≡ d (mod n) imply
a · c ≡ b · d (mod n)
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
The modulo operation can be applied to intermediate results:
(a + b) mod m ≡ (a mod m) + (b mod m) mod m
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
The modulo operation can be applied to intermediate results:
(a + b) mod m ≡ (a mod m) + (b mod m) mod m
(a · b) mod m ≡ (a mod m) · (b mod m) mod m
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
The modulo operation can be applied to intermediate results:
(a + b) mod m ≡ (a mod m) + (b mod m) mod m
(a · b) mod m ≡ (a mod m) · (b mod m) mod m
Example
83 mod 7 ≡ (8 mod 7) · (8 mod 7) · (8 mod 7) mod 7 ≡
1 · 1 · 1 mod 7 ≡ 1
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Definition
Let a, b ∈ Zn and n be a positive integer. We say a is a
multiplicative inverse of b modulo n if a · b = 1 mod n.
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Definition
Let a, b ∈ Zn and n be a positive integer. We say a is a
multiplicative inverse of b modulo n if a · b = 1 mod n.
Example
36 is a multiplicative inverse of 87 modulo 101.
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Definition
Let a, b ∈ Zn and n be a positive integer. We say a is a
multiplicative inverse of b modulo n if a · b = 1 mod n.
Example
36 is a multiplicative inverse of 87 modulo 101.
Example
The numbers 1, 2, 4, 5, 7 and 8 have multiplicative inverses in Z9 .
The numbers 0, 3 and 6 don’t have multiplicative inverses in Z9 .
Introduction to Number Theory II
Outline
Basic properties of congruence
Modular addition and multiplication
Definition
Let a, b ∈ Zn and n be a positive integer. We say a is a
multiplicative inverse of b modulo n if a · b = 1 mod n.
Example
36 is a multiplicative inverse of 87 modulo 101.
Example
The numbers 1, 2, 4, 5, 7 and 8 have multiplicative inverses in Z9 .
The numbers 0, 3 and 6 don’t have multiplicative inverses in Z9 .
Theorem
An integer a > 0 has a multiplicative inverse modulo n if and only
if gcd(a, n) = 1.
Introduction to Number Theory II