Chapter 3

1. Chapter 3 Elementary Number Theory and Methods of Proof

2. 3.3 Direct Proof and Counterexample 3 Divisibility

3. Divisibility • Definition • If n and d are integers, then n is divisible by d if, and only if, n = dk for some integers k. • The notation d|n is read “d divides n.” • Symbolically, if n and d are integers, • d|n ⇔ ∃an integer k such that n = dk.

4. Examples • Divisibility • Is 21 divisible by 3? • Yes, 21 = 3 * 7 • Does 5 divide 40? • Yes, 40 = 5 * 8 • Does 7|42? • Yes, 42 = 7 * 6 • Is 32 a multiple of -16? • Yes, 32 = (-16) * (-2) • Is 7 a factor of -7? • Yes, -7 = 7 * (-1)

5. Divisors • Divisors of zero • If d is any integer, does d divide 0? • Recall: d|n ⇔ ∃an integer k such that n = dk • n = 0, is there an integer k such that dk = 0 (n) • Yes, 0 = d*0 (0 is an integer) • Divisors of one • Which integers divide 1? • 1 = dk, 1*1 or (-1) * (-1)

6. Divisors • Positive Divisors of a Positive Number • Suppose a and b are positive integers and a|b. • Is a≤b? • Yes. a|b means that b = ka for some integer k. k must be positive because b and a are positive. • 1 ≤ k • a ≤ k * a = b, hence, a ≤ b

7. Divisibility • Algebraic expressions • 3a + 3b divisible by 3 (a & b are integers)? • Yes. By distributive law • 3a + 3b = 3(a + b), a & b are integers so the sum of a, b are integers. • 10km divisible by 5 (k & m are integers)? • 10km = 5 * (2km)

8. Prime Numbers • An alternative way to define a prime number is to say that an integer n > 1 is prime if, and only if, its only positive integer divisors are 1 and itself.

9. Transitivity of Divisibility • Prove that for all integers a, b, and c, if a|b and b|c, then a|c. • Starting point: Suppose a, b, and c are particular but arbitrarily chosen integers such that a|b and b|c. • To show: a|c • a|c, c = a*(some integers) • since a|b, b = ar for some integer r • And since b|c, c = bs • c = (ar)s(substitue for b) • c = a(rs) (assoc law) • c = ak (such that rs is integer due to close property of int) • Theorem 3.3.1 Transitivity of Divisibility

10. Divisibility by a Prime • Any integer n > 1 is divisible by a prime number. • Suppose n is a integer that is greater than 1. • If n is prime, then n is divisible by a prime number (namely itself). If n is not prime, then • n = r0s0 where r0 and s0 are integers and 1 < r0 < n and 1 < s0 < n. • it follows by definition of divisibility that r0|n. • if r0 is prime, then r0 is a prime number that divides n, and we are done. If r0 is not prime, then • r0 = r1s1 where r1 and s1 are integers and 1 < r1 < r0 • etc.

11. Counterexamples and Divisibility • Is it true or false that for all integers a and b, if a|b and b|a then a = b? • Starting Point: Suppose a and b are integers such that a|b and b|a. • b = ka & a =lb (for some integers k and l) • b = ka = k(lb) = (kl)b • factor b assuming b≠0 • 1 = kl • Thus, k = l = 1 or k = l = -1

12. Unique Factorization Theorem • Theorem 3.3.3 Fundamental Theorem of Arithmetic • Given any integer n > 1, there exist a positive integer k, distinct prime numbers p1, p2, …, pk, and positive integers e1, e2, … , ek such that • n = p1e1p2e2 p3e3 p4e4 … pkek, • and any other expression of n as a product of prime numbers is identical to this except, perhaps, for the order in which the factors are written.

13. Standard Factored Form • Given any integer n > 1, the standard factored form of n is an expression of the form • n = p1e1 p2e2 p3e3 p4e4 … pkek, • where k is a positive integer; p1, p2, … ,pk are prime numbers; e1, e2, … , ek are positive integers; and p1 < p2 < … < pk.

14. Example • Write 3,300 in standard factored form. • 3,300 = 100 * 33 = 4 * 25 * 3 * 11 • = 2 * 2 * 5 * 5 * 3 * 11 • = 22 * 52 * 3 * 11