10-19-2009 Divisibility of Natural Numbers We now return to our discussion of the natural numbers. We have built up much of the mathematical foundation for the natural numbers (N = 1; 2; 3; :::). We used set theory to come up with rigorous definition for natural numbers. We defined four standard operations on natural numbers (+; −×; ÷). We have described in much detail various conceptual models for the natural numbers and observed how these numbers are a \natu- ral' and foundational component of quantitative thinking. Now we will investigate these numbers on a deeper and more sophisticated level. The goal of this investigation is to build up the intellectual structure necessary to establish the existence and properties of more complex mathematical number systems, particularly the rational (fractions) and real numbers. Our most powerful tools in this section will turn out to be inductive and deductive reasoning. Divisibility. For any whole numbers a and b we know (by say, the repeated subtraction or division algorithm) that there are unique whole numbers q and r so that a = b · q + r where 0 ≤ r < b. Here q is called the quotient and r is called the remainder. For example, consider the problem 14 ÷ 3 = 4 R2. We can write 14 = 3 · 4 + 2: When the remainder is r = 0 in such a representation we say that b divides a. That is, if there is a unique whole number q so that a = bq then b divides a. Or in other words, we say that b divides a when b divides into a with no remainder. Warning: Never write a ÷ b when you mean to say \a divides b". To say \a divides b" you must always write out the words. Example. 7 divides 35, 25 divides 100, 341 divides 0. 8 does not divide 21, 10 does not divide 51. Example. A number a is even if 2 divides a. A number a is odd if 2 does not divide a. So we know that if a is an even whole number that a = 2 · n for some other whole number n. Also, suppose a is odd. Then a = 2n + r for some n and some 0 < r < 2. So the remainder must be 1. From this we may conclude that every odd number can be written as a = 2n + 1: We have already seen this idea before in the pattern recognition section. From now on, if we want to prove something is even or odd we will use these characterizations. Factors. If b divides a, then b is sometimes called factor or divisor of a and a is called a multiple of b. Example. Find the divisors of 8. Is 32 a multiple of 8? By guess and check, the divisors of 8 are 1,2,4,8. 32 is a multiple of 8 since 8 divides 32. Prime. A number is called prime if its only factors are 1 and itself. Composite A number is called composite if it is not prime, that is if it has more than two distinct factors. As a technical point, we do not consider the number 1, prime or composite. Prime numbers are of special importance because they are the basic building block of all numbers. Indeed, any natural number (other than 1) can be written as the product of prime numbers. For example 180 = 18 · 10 = 3 · 6 · 2 · 5 = 3 · 2 · 3 · 2 · 5 = 22 · 32 · 5: We summarize this idea in a simple but important theorem. Theorem 1. Every natural number greater than 1 is a power of a prime or can be expressed as a product of powers of primes. This factorization is unique, apart from the ordering. That is, If we ignore the order in which we write the prime factors there is only one prime factorization of every natural number. If we know the prime factorization of a number we can make an orderly list of its q1 q2 qk divisors. (See page 235-236 for two examples). To see why, suppose that a = p1 ·p2 ··· pk . If b divides a, we know that q1 q2 qk p1 · p2 ··· pk = a = b · q: This says that b must be some combination of powers of p1; p2; :::; pk and that q must be all the factors which don't appear in a. To list the divisors, we make a list of all the possible ways to write products of powers of p1; p2; :::; pk so that the product is less than or equal to a. Since primes are very important numbers, two questions quickly arise in our study of them: 1. How many primes are there? 2. How do we know that a number is prime? It turns out that there are infinitely many prime numbers. Proof. The first prime number is 2. The second prime number is 3. Consider the number 2 · 3 + 1 = 7. Since by the division algorithm, there is only one way to write 7 = 2 · q1 + r1 and 7 = 3 · q2 + r2, we discover that the number 7 is not divisible by 2 or by 3. Since every number larger than 1 has a prime factorization, there must be a prime number other than 3 which divides into 7. In fact, this number happens to be 7. Now, we build the number 2 · 3 · 7 + 1 = 43. Using the division algorithm, we can see that none of 2,3, or 7 divides 43. So there must be a new prime other than 7 which divides 43. This number happens to be 43. We continue and build the number 2 · 3 · 7 · 43 + 1 = 1807. Again, this number is not divisible by 2,3,7, or 43. So there must be a new prime other than 43 which divides 1807. In this case, we could use either the prime 13 or 139. We continue and write 2 · 3 · 7 · 13 · 43 + 1... The idea is that we can keep doing this process forever and at each step get a new prime number. So there must be an infinite number of primes. To determine if a number is prime the following theorem is very helpful. Theorem 2. If n is a composite number, then there is a prime p such that p divides n p and p ≤ n. Proof. Since n is composite we can write n = a · b where we assume that a is the smaller factor (that is a ≤ b). Now since a ≤ b multiplying by a we get a2 ≤ ab = n So p a ≤ n: Now a is a whole number bigger than 1, and so has a prime factorization. If p is a prime in this factorization then p ≤ a is a divisor of a and it is also a divisor of b. p So there is a prime p ≤ n which divides n. p Consequently, by the rule of indirect reasoning, if there is no prime p ≤ n which divides n then n is prime. (Why?) For examples of using this theorem, see 237-239..