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Lecture 6 Integers

Lecture 6 Integers. CSCI – 1900 Mathematics for Computer Science Spring 2014 Bill Pine. Lecture Introduction. Reading Kolman Section 1.4 Remainder Theorem Divisibility of integers Prime numbers GCD LCM Representing integers in different bases. Remainder Theorem.

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Lecture 6 Integers

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  1. Lecture 6Integers CSCI – 1900 Mathematics for Computer Science Spring 2014 Bill Pine

  2. Lecture Introduction • Reading • Kolman Section 1.4 • Remainder Theorem • Divisibility of integers • Prime numbers • GCD • LCM • Representing integers in different bases CSCI 1900

  3. Remainder Theorem • Given two integers, n and m with n> 0, • Perform the integer division of m by n • q is the quotient and r is the remainder • q and r are unique because we require 0 <= r< n • Therefore, we can write m=q*n + r • This result is known as the Remainder Theorem CSCI 1900

  4. Examples of m = qn + r • If n is 3 and m is 16 • 16 = 5*3 + 1 q = 5; r = 1 • If n is 10 and m is 3 • 3 = 0*10 + 3 q = 0; r = 3 • If n is 5 and m is –11 • 11 = – 3*5 + 4 q = – 3; r = 4 CSCI 1900

  5. Divisibility • If one integer, n, divides into a second integer, m, without a remainder, then we say that • n divides m • Denoted n | m • If one integer, n, does not divide evenly into a second integer, m, then we say that • n does not divide m • Denoted n | m CSCI 1900

  6. Some Properties of Divisibility • If n | m, • There exists an integer k such that m = k * n • The absolute values of both k and n are less than the absolute value of m, i.e., |n| < |m| and |k| < |m| • Examples: 4 | 24 24 = 4 * 6 both 4 and 6 are less than 24 5 | 135 135 = 5 * 27 both 5 and 27 are less than 135 CSCI 1900

  7. Simple properties of divisibility • Given three integers a, b, c with a | b and a | c, then • a | (b + c) • a | (b - c) • a | bc • Given three integers a, b, c with a | b and b | c, then • a | c CSCI 1900

  8. Prime Numbers • A number p is called prime if the only positive integers that divide p are p and 1 • Examples of prime numbers: 2, 3, 5, 7, 11 • There are many computer algorithms that can be used to determine if a number n>1 is prime, with greater or lesser efficiency • Who cares ? • Anyone who buys anything online or has a wireless network they do not want to share ! • Cryptography involves prime number in some manner CSCI 1900

  9. Basic Prime Number Algorithm Function IsPrime( n ) nIsPrime = True for i= 2 to n-1 if( i | n) nIsPrime = False Exit Loop return (nIsPrime) CSCI 1900

  10. Factoring a Number into its Primes • Repeatedly dividing a number into its multiples until the multiples no longer can be divided, shows us that any number can be expressed a a product prime numbers • Examples: 9 = 3 * 3 = 3224 = 8 * 3 = 2 * 2 * 2 * 3 = 23 * 3315 = 3*105 = 3*3*35 = 3*3*5*7 = 32 * 5 * 7 • Any number can be expressed as a product of prime numbers • This factorization is unique CSCI 1900

  11. Modulus • The mod n operator is a direct consequence of the Remainder Theorem • m mod n is defined to be the remainder when m is divided by n • The divisor n is called the modulus • Given m = q * n + r then we say m mod n = r • If m mod n = 0 then m | n CSCI 1900

  12. Modulus(cont) • Examples 13 mod 3 = 1 => 4*3+1=13 32 mod 5 = 2 => 6*5+2=32 a mod 7 = 1 => 7*k+1=a, for some integer k CSCI 1900

  13. Greatest Common Divisor • If a, b, and c are in Z+, and c | a and c | b, we say that c is a common divisor of a and b • If d is the largest such c, d is called the greatest common divisor (GCD) • d is a multiple of every c, i.e., every c divides d • If the GCD(a, b) = 1 then a and b are relatively prime CSCI 1900

  14. GCD Example Find the GCD of 540 and 315: • First find the prime factors of each 540 = 22* 33 * 5 315 = 32 *5* 7 • 540 and 315 share the divisors 3 and 5, 540 has 33 and 5 315 has 32 and 5 • So each is equal 32 * 5 times some different primes So the largest is the GCD  32 * 5 = 45 • 315  45 = 7 and 540  45=12 CSCI 1900

  15. Euclid’s Algorithm • Inputs: two positive integers a and b, a > b • Output: gcd(a, b) – the greatest common divisor of a and b • Procedure: r=amodb while ( r> 0 ) a=b b=r r=amodb returnb CSCI 1900

  16. Euclid’s Algorithm Example • For two integers a= 846 and b = 212 846 = 3 * 212 + 210 k1 = 3; r1 = 210 212 = 1 * 210 + 2 k2 = 1; r2 = 2 210 = 105 * 2 + 0 k3 = 105; r3 = 0 GCD=2 • For two integers a= 555 and b = 296 555 = 1 * 296 + 259 k1 = 1; r1 = 259 296 = 1 * 259 + 37 k2 = 1; r2 = 37 259 = 7 * 37 + 0 k3 = 7; r3 = 0 GCD = 37 846 = 47 * 32 * 2 212 = 53 * 22 555 = 37 * 5 * 3 296 = 37 * 23 CSCI 1900

  17. Least Common Multiple • If a, b, and k are in Z+, and a | k, b | k, we say that k is a common multiple of a and b • The smallest such k, call it c, is called the least common multiple or LCM of a and b • We write c = LCM(a, b) • An important result is • GCD(a, b)*LCM(a, b) = a*b • This provides a convenient way to calculate LCM(a, b) CSCI 1900

  18. Representation of Integers • In day-to-day life, we use decimal (base 10) arithmetic , but it is only one of many ways to express an integer value • We say that a decimal value is the “base 10 expansion of n”or the “decimal expansion of n” • If b > 1 is an integer, then every positive integer n can be uniquely expressed in the form:n = dkbk + dk-1bk-1 + dk-2bk-2 + … + d1b1 + d0b0where 0 < di< b, i = 0, 1, …, k CSCI 1900

  19. Algorithm: Base 10 to Base b • Input: two positive integers, base b and number n in base 10 • Output: the value of n in base b • Procedure: See Handout CSCI 1900

  20. Example: Decimal 482 to Base 5 482 = 96*5 + 2 (remainder (2) is d0 digit) 96 = 19*5 + 1 (remainder (1) is d1 digit) 19 = 3*5 + 4 (remainder (4) is d2 digit) 3 = 0*5 + 3 (remainder (3) is d3 digit) 48210 = 34125 CSCI 1900

  21. Example: Decimal 704 to Base 8 (Octal) • 704 = 88*8 + 0 (remainder (0) is d0 digit) • 88 = 11*8 + 0 (remainder (0) is d1 digit) • 11 = 1*8 + 3 (remainder (3) is d2 digit) • 1 = 0*8 + 1 (remainder (1) is d3 digit) • 70410 = 13008 CSCI 1900

  22. Algorithm: Base b to Base 10 • Input: two positive integers, base b and number n in base b • Output: the value of n in base 10 • Procedure: See Handout CSCI 1900

  23. Example: 32125 to Base 10 34125 = 3 * 53 + 4 * 52 + 1 * 51 + 2 * 50 = 3 * 125 + 4 * 25 + 1 * 5 + 2 * 1 = 375 + 100 + 5 + 2 = 48210 CSCI 1900

  24. Example: 13008 to Base 10 13008= 1 * 83 + 3 * 82 + 0 * 81 + 0 * 80 = 1 * 512 + 3 * 64 + 0 * 8 + 0 * 1 = 512 + 192 = 70410 CSCI 1900

  25. Nota Bene • The two conversion algorithms are pairs • If you convert a number n from base 10 to base b • You can check your result by converting the result back to base 10 • If you convert a number n from base b to base 10 • You can check your result by converting the result back to base b CSCI 1900

  26. Key Concepts Summary • Divisibility of integers • Prime numbers • Remainder Theorem • GCD • LCM • Expansion into different base CSCI 1900

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