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82 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 diāµerent 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).