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Section 2.4. The Integers and Division. Number Theory. Branch of mathematics that includes (among other things): divisibility greatest common divisor modular arithmetic. Division. Division of one integer by another (e.g a/b) produces 2 results: quotient: number of time b “goes into” a I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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1. Section 2.4 The Integers and Division

2. Number Theory • Branch of mathematics that includes (among other things): • divisibility • greatest common divisor • modular arithmetic

3. Division • Division of one integer by another (e.g a/b) produces 2 results: • quotient: number of time b “goes into” a • remainder: what’s left over if the values don’t divide evenly

4. Division • If a and b are integers and a  0, a divides b if there exists an integer c such that b = ac • This also means that c divides b • a and c are factors of b • b is a multiple of both a and c • The notation a|b means a divides, or is a factor, of b

5. Division • If n and d are integers, how many positive integers <= n are divisible by d? • All integers divisible by d are of the form dk (where k is a positive integer) • So the positive integers divisible by d which are <= n are is the set of all k’s such that: • 0 < dk <= n or 0 < k <= n/d • Thus, there are n/d positive integers <= n which are divisible by d

6. Theorem 1 • Let a, b & c be integers. Then: • if a|b and a|c, then a|(b+c) • if a|b then a|bc, for all integers c • if a|b and b|c, then a|c

7. Proof of Theorem 1 • Part 1: if a|b and a|c, then a|(b+c) • If a|b and a|c, there must be integers s & t such that b = as and c = at • So b+c = as+at = a(s+t) • Then by definition of divisibility, a|(b+c) • Part 2: if a|b then a|bc for all integers c • If a|b, then b = at for some integer t • so bc = a(tc) and, by definition, a|bc

8. Prime Numbers • A positive integer that has only 2 positive integer factors (1 and itself) is a prime number • A positive integer > 1 that is not prime is a composite

9. Theorem 2: Fundamental Theorem of Arithmetic • Every positive integer can be written as the product of primes • Usually, the prime factors are written in increasing order, for example: 2 x 3 x 103 = 618

10. Theorem 3 • If n is a composite integer, then n has a prime divisor <= n • For example, 103 is prime because: 103 = 10 and the primes < 10 are 2, 3, 5 and 7 • Since none of these is a factor of 103, 103 must be prime

11. Procedure for determining prime factors of and integer n • Divide n by successive primes, beginning with 2 • If n has a prime factor, then some prime number p <= n will be found divisible by n • If such a value p is found, continue by factoring n/p • look for value q such that p < q <= n/p • if found, continue by factoring n/pq, etc.

12. Example Find prime factorization of 65238: 65238 = 2 x 32619 = 2 x 3 x 10873 Testing prime numbers: 5, 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79 Finally, a factor is found: 65238 = 2 x 3 x 83 x 131 Since 131 < 83, no further testing required - 131 is prime

13. Theorem 4: the Division Algorithm • For any integer a and positive integer d, there exist unique integers q and r such that: a = dq + r with 0 <= r <= d • In the above expression: • a is the dividend • d is the divisor • q is the quotient • r is the remainder (always positive)

14. Greatest Common Divisors • For two non-zero integers a and b, the largest integer d such that d|a and d|b is the greatest common divisor of a & b, denoted gcd (a,b)

15. Finding gcd: method 1 • Find all possible divisors of both numbers, and choose the largest one they have in common • Example: find gcd(81, 99) • factors of 81: 1, 3, 9, 27, 81 • factors of 99: 1, 3, 9, 11, 33, 99 • so gcd(81, 99) = 9

16. Relatively prime numbers • Two numbers are relatively prime if their gcd is 1 • Integers in a set {a1, a2, … an} are pairwise relatively prime if: gcd(ai,aj) = 1 whenever 1 <= i <= j <=n

17. Relative prime examples • (14,15,21): gcd(14,15) = 1 gcd(14, 21) = 7 gcd(15,21) = 3 so they are not relatively prime • (7,8,9,11) gcd(7,8) = 1 gcd(8,9) = 1 gcd(9,11) = 1 gcd(7,9) = 1 gcd(8,11) = 1 gcd(7,11) = 1 so they are relatively prime

18. Method 2 for finding gcd • Use prime factorizations of integers: a = p1a1*p2a2* … *pnan b = p1b1*p2b2* … *pnbn • each exponent is non-negative • all primes occurring in the factorizations of either a or b are included in both factorizations, with 0 exponents where necessary • gcd(a,b) = p1min(a1,b1)*p2min(a2,b2)*…*pnmin(an,bn)

19. Example a = 12, b = 9 12 = 21 * 21 * 31 * 30 =22 *31 9 = 20 * 20 * 31 * 31 = 20 * 32 So gcd(12,9) = 2min(0,2) *3min(1,2) = 20 * 31 = 3

20. Least Common Multiple • For two positive integers a and b, the lcm(a,b) is the smallest positive integer that is divisible by both a and b • In other words, lcm(a,b)=p1max(a1,b1)*p2max(a2b2)*…*pnmax(an,bn) • For example: lcm(12,9) = 2max(0,2) * 3max(1,2) = 22 * 32 = 36

21. Theorem 5 • For positive integers a and b, the product of a and b is equal to gcd(a,b) * lcm(a,b) • For example, if a=12 and b=9: 12 * 9 = 108 gcd(12,9) = 3 and lcm(12,9) = 36 3 * 36 = 108

22. Modular Arithmetic • Modulus: operation that finds the remainder when one positive integer is divided by another • a mod m = r when: a = qm + r and 0 <= r < m

23. Congruence • For two integers a and b, and positive integer m, a is congruent to b mod m if m|(a-b) This congruence is denoted: a  b (mod m) a  b (mod m) if and only if a mod m = b mod m Therefore congruence occurs between a and b (mod m) if both a and b have the same remainder when divided by m

24. Congruence Examples • Determine if 80 is congruent to 5 modulo 17 • Translation: divide 80 by 17 and see if the remainder is 5 • It isn’t: 17 goes into 80 4 times, with a remainder of 12 • Is -29 congruent to 5(mod 17)? • 29 = 17 * (- 2) + 5 so -29  5 (mod 17)

25. Theorems 6 & 7 • Theorem 6: • Let m be a positive integer: • Integers a and b are congruent modulo m if and only if there is an integer k such that a = b + km • Theorem 7: • 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)

26. Section 2.4 The Integers and Division - ends -