Download 3 - Sophia Smith

1
Sec 3.4 The Integers and Division
Definition If a and b are integers with a 6= 0, we say that a divides b if there is an integer k
such that b = ak. When a divides b, we say that a is a factor of b and that b is a multiple of a. The
notation a | b denotes that a divides b. We write a - b when a does not divide b.
Example 1 3 | 12 and 4 - 7.
Example 2 Let n and d be positive integers. How many positive integers not exceeding n are
divisible by d?
Example 3 Suppose a | b and b | a. What can you say about a and b?
Theorem 1 Let a, b, and c are integers. Then
1. if a | b and a | c, then a | b + c.
2. if a | b, then a | nb for all integers n.
3. if a | b and b | c, then a | c.
Corollary If a, b, and c are integers such that a | b and a | c, then a | mb + nc.
Theorem 2 (Division Algorithm) Let a be an integer and d a positive integer. Then there are
unique integers q and r, with 0 ≤ r < d, such that a = dq + r. In this case, we say that q is the
quotient and r is the remainder.
Example 4 Verify Theorem 2 above using (a, d) = (10, 3), (−17, 5), and (60, 12).
Definition If a and b are integers and m is a positive integer, then a is said to be congruent to
b modulo m if m divides a − b. We use the notation a ≡ b (mod m) to indicate that a is congruent
to b modulo m. If a and b are not congruent modulo m, we write a 6≡ b (mod m).
Example 5 12 ≡ 5 (mod 6) but 24 6≡ 14 (mod 6).
Example 6 November 1, 2010 is Monday and therefore we know that November 15, 2010 is also
Monday. Why?
Theorem 3 Let m be a positive integer. If a ≡ b (mod m) and c ≡ d (mod m), then
a + c ≡ b + d (mod m) and ac ≡ bd (mod m).
Proof
2
Sec 3.5 Primes and Greatest Common Divisors
Definition A positive integer p greater than 1 is called prime if the only positive factors of p are 1
and p. A positive integer that is greater than 1 and is not prime is called composite.
Theorem 1 (The Fundamental Theorem of Arithmetic) Every positive integer greater than 1 can
be written uniquely as a prime or as the product of two or more primes where the prime factors are
written in order of nondecreasing size.
Example 1 180 = 6 · 30 = (2 · 3) · (3 · 10) = (2 · 3) · (3 · 2 · 5) = 22 · 32 · 5. The last expression is called the prime factorization of 180.
Example 2 Find the prime factorization of 10! = 1 · 2 · 3 · · · · · 10.
Example 3 Let 2a · 3b · 5c · · · · be the prime factorization of 100!.
1. Determine a, b, and c.
2. How many zeros does 100! have in the end?
Determining whether given number is prime is important in many applications(such as cryptography), and the following theorem provides a useful criterion.
Theorem
2 If n is a composite integer, then n must have a prime factor less than or equal to
√
n.
Example 4 Show that 101 is prime.
3
Theorem 3 There are infinitely many primes.
Some Open Questions about Primes
1. Goldbach’s Conjecture Can every even integer greater than 2 be written as the sum of two
primes? It is true up to 2 × 1017 .
2. Twin Prime Conjecture Twin primes are primes that differ by 2, such as 3 and 5, 5 and 7,
11 and 13, and 17 and 19. Do we have infinitely many twin primes? The largest twin primes
known so far are
16, 869, 987, 339, 975 × 2171,960 ± 1.
Definition Let a and b be integers, not both zero. The largest integer d such that d | a and d | b is
called the greatest common divisor of a and b. The greatest common divisor of a and b is denoted
by gcd(a, b). The integers a and b are relatively prime (or coprime) if gcd(a, b) = 1.
Example 5 gcd(24, 36) =
.
Definition The least common multiple of the positive integers a and b is the smallest positive integer
that is divisible by both a and b. The least common multiple of a and b is denoted by lcm(a, b).
Example 6 What is lcm(23 · 35 · 72 , 24 · 33 )?
Remark Let pk denote the kth prime number so that p1 = 2, p2 = 3, p3 = 5, and so on.
1. By the Fundamental Theorem of Arithmetic, for every positive integer a, we can find an n and
natural numbers ek ’s such that
a = pe11 pe22 pe33 · · · penn .
2. a = pe11 pe22 pe33 · · · penn and b = pf11 pf22 pf33 · · · pfmm be positive integers. Taking the zeroth power if
necessary, we can assume that m = n. Then
min{e1 ,f1 } min{e2 ,f2 } min{e3 ,f3 }
p2
p3
gcd(a, b) = p1
· · · pnmin{en ,fn }
and
max{e1 ,f1 } max{e2 ,f2 } max{e3 ,f3 }
p2
p3
lcm(a, b) = p1
3. In particular, we get ab = gcd(a, b) · lcm(a, b).
n ,fn }
· · · pmax{e
.
n