Download 5.2. Counting Factors, Greatest Common Factor, and Least

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5. NUMBER THEORY
5.2. Counting Factors, Greatest Common Factor, and Least
Common Multiple
Counting Factors
Example. How many factors (or divisors) does 10,800 have?
First, 10, 800 = 24 · 33 · 52 as a product of primes.
Then a divisor of 10,800 can only have 2, 3, and 5 as prime factors, with no
more than 4 twos, no more than 3 threes, and no more than 2 fives.
There are five possibilities for the number of twos to include: 0, 1, 2, 3, 4.
There are four possibilities for the number of threes to include: 0, 1, 2, 3.
There are three possibilities for the number of fives to include: 0, 1, 2.
Thus there are 5 · 4 · 3 = 60 di↵erent factors of 10,800.
Theorem. Suppose that a counting number n is expressed as a product
of distinct primes with their respective exponents, say
n = (pn1 1 )(pn2 2 ) · · · (pnmm ).
Then the number of factors of n is the product
(n1 + 1)(n2 + 1) · · · (nm + 1).
Greatest Common Factor
Definition (Greatest Common Factor).
The greatest common factor (GCF) of two (or more) nonzero whole numbers
is the largest whole number that is a factor of both (all) of the numbers. The
GCF of a and b is written GCF(a, b).