Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers.

We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:

N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbers

Z for the set of integers.Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z.

We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Natural numbers and integers

Positive integers are also called natural numbers. We write:N for the set of natural numbersZ for the set of integers.

Definition: Suppose d , n ∈ Z. We say d divides n if there is k ∈ Z such thatdk = n.

Alternative phrasing:

I n is divisible by d

I d is a divisor of n

I d is a factor of n

I n is a multiple of d .

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”.

Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:

I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12

I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12

I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every n

I d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every d

I 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.

6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6.

30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.

So 30 = (5× 2)× 3, so 3 | 30.

Divisors

We write d | n to mean “d divides n”. Do not confuse d | n with d/n!

Examples:I 3 | 12I −3 | 12I 1 | n for every n

I n | n for every nI d | 0 for every dI 0 | n if and only if n = 0.

Lemma 4.1: Suppose a, b, c ∈ Z. If a | b and b | c then a | c.

e.g. a = 3, b = 6, c = 30.6 = 2× 3, so 3 | 6. 30 = 5× 6, so 6 | 30.So 30 = (5× 2)× 3, so 3 | 30.

Primes

The positive factors of 20 are1, 2, 4, 5, 10, 20.

The positive factors of 19 are1, 19.

Definition: Suppose n ∈ N.n is prime if n > 1 and the only positive factors of n are 1 and nn is composite if n > 1 and n is not prime.

(1 is neither prime nor composite.)

The primes are2, 3, 5, 7, 11, 13, 17, 19, 23, . . .

Primes

The positive factors of 20 are1, 2, 4, 5, 10, 20.

The positive factors of 19 are1, 19.

Definition: Suppose n ∈ N.n is prime if n > 1 and the only positive factors of n are 1 and nn is composite if n > 1 and n is not prime.

(1 is neither prime nor composite.)

The primes are2, 3, 5, 7, 11, 13, 17, 19, 23, . . .

Primes

The positive factors of 20 are1, 2, 4, 5, 10, 20.

The positive factors of 19 are1, 19.

Definition: Suppose n ∈ N.n is prime if n > 1 and the only positive factors of n are 1 and nn is composite if n > 1 and n is not prime.

(1 is neither prime nor composite.)

The primes are2, 3, 5, 7, 11, 13, 17, 19, 23, . . .

Primes

The positive factors of 20 are1, 2, 4, 5, 10, 20.

The positive factors of 19 are1, 19.

Definition: Suppose n ∈ N.

n is prime if n > 1 and the only positive factors of n are 1 and nn is composite if n > 1 and n is not prime.

(1 is neither prime nor composite.)

The primes are2, 3, 5, 7, 11, 13, 17, 19, 23, . . .

Primes

The positive factors of 20 are1, 2, 4, 5, 10, 20.

The positive factors of 19 are1, 19.

Definition: Suppose n ∈ N.n is prime if n > 1 and the only positive factors of n are 1 and n

n is composite if n > 1 and n is not prime.(1 is neither prime nor composite.)

The primes are2, 3, 5, 7, 11, 13, 17, 19, 23, . . .

Primes

The positive factors of 20 are1, 2, 4, 5, 10, 20.

The positive factors of 19 are1, 19.

Definition: Suppose n ∈ N.n is prime if n > 1 and the only positive factors of n are 1 and nn is composite if n > 1 and n is not prime.

(1 is neither prime nor composite.)

The primes are2, 3, 5, 7, 11, 13, 17, 19, 23, . . .

Primes

The positive factors of 20 are1, 2, 4, 5, 10, 20.

The positive factors of 19 are1, 19.

Definition: Suppose n ∈ N.n is prime if n > 1 and the only positive factors of n are 1 and nn is composite if n > 1 and n is not prime.

(1 is neither prime nor composite.)

The primes are2, 3, 5, 7, 11, 13, 17, 19, 23, . . .

Primes

The positive factors of 20 are1, 2, 4, 5, 10, 20.

The positive factors of 19 are1, 19.

Definition: Suppose n ∈ N.n is prime if n > 1 and the only positive factors of n are 1 and nn is composite if n > 1 and n is not prime.

(1 is neither prime nor composite.)

The primes are2, 3, 5, 7, 11, 13, 17, 19, 23, . . .

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n. X

If n is not prime, then n has a factor a, where 1 < a < n. By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n. So p is a prime factor of n. X

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n. X

If n is not prime, then n has a factor a, where 1 < a < n. By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n. So p is a prime factor of n. X

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n. X

If n is not prime, then n has a factor a, where 1 < a < n. By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n. So p is a prime factor of n. X

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n.

X

If n is not prime, then n has a factor a, where 1 < a < n. By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n. So p is a prime factor of n. X

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n. X

If n is not prime, then n has a factor a, where 1 < a < n. By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n. So p is a prime factor of n. X

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n. X

If n is not prime, then n has a factor a, where 1 < a < n.

By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n. So p is a prime factor of n. X

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n. X

If n is not prime, then n has a factor a, where 1 < a < n. By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n. So p is a prime factor of n. X

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n. X

If n is not prime, then n has a factor a, where 1 < a < n. By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n.

So p is a prime factor of n. X

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n. X

If n is not prime, then n has a factor a, where 1 < a < n. By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n. So p is a prime factor of n.

X

Prime factors

Lemma 4.2: If n > 1, then n has at least one prime factor.

Idea of proof: Use strong induction.

If n is prime, then n is a prime factor of n. X

If n is not prime, then n has a factor a, where 1 < a < n. By the inductivehypothesis a has a prime factor p.

Now p | a and a | n, so p | n. So p is a prime factor of n. X