Download Primes, Greatest Common Divisors, and Least Common Multiple in Discrete Mathematics I and more Study Guides, Projects, Research Elementary Mathematics in PDF only on Docsity!
4.3 Primes and Greatest Common Divisors
Primes
An integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite.
The Fundamental Theory of Arithmetic
Every integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are written in order of nondecreasing size.
Theorem 2
If n is a composite integer, then n has a prime divisor less than or equal to
n.
Greatest Common Divisor
Let a and b be integers, not both zero. The largest integer d such that d | a and d | b is called the greatest common divisor of a and b. The greatest common divisor of a and b is denoted by gcd(a, b).
Finding the Greatest Common Divisor using Prime Factorization
Suppose the prime factorizations of a and b are:
a = pa 11 pa 22 · · · pa nn
b = pb 11 pb 22 · · · pb nn
where each exponent is a nonnegative integer, and where all primes occurring in either prime factorization are included in both, with zero exponents if necessary. Then:
gcd(a, b) = pmin 1 (a^1 ,b^1 )pmin 2 (a^2 ,b^2 )· · · pmin n (an,bn)
Relatively Prime
The integers a and b are relatively prime if their greatest common divisor is 1.
Least Common Multiple
The least common multiple of the positive integers a and b is the smallest positive integer that is divisible by both a and b. The least common multiple of a and b is denoted by lcm(a, b).
Finding the Least Common Multiple Using Prime Factorizations
Suppose the prime factorizations of a and b are:
a = pa 11 pa 22 · · · pa nn
b = pb 11 pb 22 · · · pb nn
where each exponent is a nonnegative integer, and where all primes occurring in either prime factorization are included in both, with zero exponents if necessary. Then:
lcm(a, b) = pmax 1 (a^1 ,b^1 )pmax 2 (a^2 ,b^2 )· · · pmax n (an,bn)
Theorem 5
Let a and b be positive integers. Then ab = gcd(a, b) · lcm(a, b).
The Euclidean Algorithm
Let a = bq + r where a, b, q, and r are integers. Then gcd(a, b) = gcd(b, r). Also written as gcd(a, b) = gcd((b, (a mod b)).
4.3 pg 272 # 3
Find the prime factorization of each of these integers.
a) 88 √ 88 ≈ 9. 38 88 /2 = 44 44 /2 = 22 22 /2 = 11 Therefore 88 = 2^3 · 11
b) 126 √ 126 ≈ 11. 22 126 /2 = 63 63 /3 = 21 21 /3 = 7 Therefore 63 = 2 · 32 · 7
c) 729 √ 729 = 27 729 /3 = 243 243 /3 = 81 81 /3 = 27 27 /3 = 9 9 /3 = 3 Therefore 729 = 3^6
4.3 pg 273 # 29
Find gcd(92928, 123552) and lcm(92928, 123552) and verify that gcd(92928, 123552)·lcm(92928, 123552) = 92928 · 123552. [Hint: First find the prime factorizations of 92928 and 123552.]
92928 = 2^8 · 3 · 112 123552 = 2^5 · 33 · 11 · 13
gcd(92928, 123552) = 2^5 · 3 · 11 lcm(92928, 123552) = 2^8 · 33 · 112 · 13
gcd(92928, 123552) · lcm(92928, 123552) = 92928 · 123552 (2^5 · 3 · 11) · (2^8 · 33 · 112 · 13) = (2^8 · 3 · 112 ) · (2^5 · 33 · 11 · 13) 213 · 34 · 113 · 13 = 2^13 · 34 · 113 · 13
4.3 pg 273 # 33
Use the Euclidean algorithm to find
c) gcd(1001, 1331) 1331 = 1001 · 1 + 330 1001 = 330 · 3 + 11 330 = 11 · 30 + 0 gcd(1001, 1331) = gcd(1001, 330) = gcd(330, 11) = gcd(11, 0) = 11
f) gcd(9888, 6060) 9888 = 6060 · 1 + 3828 6060 = 3828 · 1 + 2232 3828 = 2232 · 1 + 1596 2232 = 1596 · 1 + 636 1596 = 636 · 2 + 324 636 = 324 · 1 + 312 324 = 312 · 1 + 12 312 = 12 · 26 + 0 gcd(9888, 6060) = gcd(6060, 3820) = gcd(3828, 2232) = gcd(2232, 1596) = gcd(1596, 636) = gcd(636, 324) = gcd(324, 312) = gcd(32, 12) = gcd(12, 0) = 12
