120=2×60=2×(2×30)=2×2×(2×15)=2×2×2×3×5.
We say that theprimefactorization of 120 is 23×3×5. We could,however, have came to this by another route. For instance
120=12×10=(3×4)×(2×5)=(3×(2×2))×(2×5)
but rearranging the prime factors from least to greatest still yields the same result as before:120=23×3×5.
At least it did in that example, and this behaviour maybe more or less familiar to you, but how can you be sure that this applies to every number? It is clear enough that any number can be broken down into a product of primes but, since there is in general more than one way of tackling this task, how can we be sure that the process will always deliver the same final result? This is an important question, so I will take a few moments to give an outline of the reasoning that allows us to be absolutely sure about this. It is a consequence of another special property of prime numbers that we shall call the euclideanproperty:ifa prime number is a factor of a product of two or more numbers, then it is a factor of one of the numbers in that product. For example,7 is a factor of 8×35=280(as the product 280=7×40)and we note that 7 is a factor of35. This property characterizes primes as no composite number can give you the same guarantee:for example,we seethat 6 is afactor of 8×15=120(as 120=6×20)yet 6 is not a factor of either 8 or 15.