Number theory has fascinated mathematicians for years. Fundamental to number theory are numbers themselves, and the basic building blocks for numbers are prime numbers. A prime number is a counting number that only has two factors, itself and one. Counting numbers which have more than two factors (such as six, whose factors are 1, 2, 3 and 6), are said to be composite numbers. The number one only has one factor and is considered to be neither prime nor composite.
When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. For example, the number 72 can be written as a product of primes as: 72 = 23• 32. The expression "23 • 32" is said to be the prime factorization of 72. The Fundamental Theorem of Arithmetic states that every composite number can be factored uniquely (except for the order of the factors) into a product of prime factors. What this means is that how you choose to factor a number into prime factors makes no difference. When you are done, the prime factorizations are essentially the same. Examine the two factor trees for 72 given below.
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