2. Divisibility October 8, 2014 1

3. Divisibility If a and b are integers and a  0, then the statement that a divides b means that there is an integer c such that b = ac. The notation is a | b. If a does not divide b, we write a | b. Note that every integer divides 0, 1 divides every integer and every integer divides itself. 2

4. Divisibility For all integers a, b and c, if c | a and c | b, then c | (a + b). Why? For all integers a, b and c, if a | b, then a | bc. Why? For all integers a, b and c, if a | b and b | c, then a | c. Why? 3 a = cx; b = cy Then, a + b = c(x + y) So, c divides (a + b)

5. Division Theorem Let a be an integer and d be a positive integer. Then, there exists unique integers q and r with 0  r < d such that a = dq + r. r is called the remainder of a divided by d and q is called the quotient. 4

6. Greatest Common Divisor Suppose that a and b are integers and not both zero. The largest integer d such that d | a and d | b is called the greatest common divisor of a and b. We write d = gcd (a, b) If gcd(a, b) = 1, then a and b are relative prime of each other. 5

7. Properties of gcd Some properties of gcd: gcd(a, b) = gcd (b, a) gcd(a, b) = gcd (-a, b) gcd(a, b) = gcd (|a|, |b|) gcd(a, 0) = |a| gcd(a, ka) = |a| where k is an integer Why each of the above? 6

8. Properties of gcd If a = bq + r for integers a, b, q and r, then gcd(a, b) = gcd(b, r) Why? (Hint: Try to show that every common divisor of a and b is also a common divisor of b and r and vice versa.) 7 Let x = cd(a, b) a = xh; b = xk Then, r = a – bq = x(h – kq) So, x | r and x = cd(a, r) Let y = cd(b, r) b = ys; r = yt Then, a = bq + r = y(qs + t) So, y | a and y = cd(a, b)

9. Euclid’s Algorithm gcd(a, b) ‏if a = 0 return |b| ‏if b = 0 return |a| ‏repeat ‏r := a mod b ‏a := b ‏b := r ‏until r = 0 ‏return a Why is this algorithm correct (i.e., will terminate and give a correct answer)? (Try to show that when a or b is 0, it is correct. Otherwise, the steps taken in the repeat- until loop reduce the value of b and keep the gcd between a and b invariant.) 8

10. GCD Identity If gcd(a, b) = d, then there exists integers m and n such that d = am + bn. That is, the gcd of two numbers can be written as a linear combination of the two numbers. This is found by a French mathematician called É tienne Bézout. Thus, this is also called the Bézout’s Identity. 9

11. A Game on Water Jugs Suppose you are given two jugs, one of 5 gallons and one 3 gallons. Now you are standing beside a fountain and is asked to fill one of the jugs with exactly 4 gallons of water. Can you do it? If so, how? How about making 9 gallons of water from 2 jugs of 10 gallons and 14 gallons? Can you do it? How or why? 10 Any volume one can measure using the 2 jugs is a linear combination of the sizes of the 2 jugs. E.g., here, any volume measurable x can be written as 10m + 14n. Since the gcd(10, 14) = 2, x must also be divisible by 2. But now, we want to measure 9. So, this is impossible.

12. Prime Numbers An integer n > 1 is a prime number means that if n = rs for positive integers r and s, then r = 1 or s = 1. That is, the only divisors of n are 1 and itself. 2, 3, 5, 7, …, 19, …, 2 57,885,161 -1, … Discovered in 2013. This number has 17,425,170 digits. Till January 2014, it’s still the largest one found. 11

13. Fundamental Theorem of Arithmetic Every positive integer greater than 1 can be written uniquely as the product of primes. That is, any positive integer n can be written as p 1 e 1 p 2 e 2 … p k e k For example, 540 =2 2  3 3  5 12

14. Composite Numbers An integer n > 1 is composite means that n is not a prime. Theorem: If p is a prime and p|ab, then p|a or p|b. Theorem: If p is a prime and p|a 1 a 2 …a n, then p|a i for some i. 13

15. Modulus Operations If a and m are integers with m > 0, then the remainder of a divided by m is denoted by a mod m. That is, when a = qm + r for 0  r < m, then r = a mod m. Easy to see that the only possible values for a mod m for any integer a is 0, 1, …, m-1. a -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 a mod 7 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 14

16. Modulus Operations For any integers a and b and positive integer m, if a mod m = b mod m, we say that a is congruent to b modulo m, or a and b are in the same congruence class. We can also write a  b mod m. This is equivalent to the fact that m divides a-b. Why? 15

17. Modulus Operations a  a (mod m) If a  b (mod m), then b  a (mod m) If a  b (mod m) and b  c (mod m), then a  c (mod m) a  b (mod m) iff a + k  b + k (mod m) If a  b (mod m), then ak  bk (mod m) If a  b (mod m) and c  d (mod m), then a + c  b + d (mod m) 16