Elements of Algebra

12 ELEMENTS SECT. I. 30 by $5 \times 6$, it is evident that $6$ not being a prime number, but being produced by $2 \times 3$, we might have represented $30$ by $5 \times 2 \times 3$, or by $2 \times 3 \times 5$; that is to say, by factors which are all prime numbers. 42. If we now consider those composite numbers which may be resolved into prime factors, we shall observe a great difference among them; thus we shall find that some have only two factors, that others have three, and others a still greater number. We have already seen, for example, that 4 is the same as $2 \times 2$, | 6 is the same as $2 \times 3$, 8 - - - $2 \times 2 \times 2$, | 9 - - - $3 \times 3$, 10 - - - $2 \times 5$, | 12 - - - $2 \times 3 \times 2$, 14 - - - $2 \times 7$, | 15 - - - $3 \times 5$, 16 - - $2 \times 2 \times 2 \times 2$, | and so on. 43. Hence, it is easy to find a method for analysing any number, or resolving it into its simple factors. Let there be proposed, for instance, the number $360$; we shall represent it first by $2 \times 180$. Now $180$ is equal to $2 \times 90$, and $\left. \begin{array}{r} 90 \\ 45 \\ 15 \end{array} \right\}$ is the same as $\left\{ \begin{array}{l} 2 \times 45, \\ 3 \times 15, \text{ and lastly} \\ 3 \times 5. \end{array} \right.$ So that the number $360$ may be represented by these simple factors, $2 \times 2 \times 2 \times 3 \times 3 \times 5$; since all these numbers multiplied together produce $360$*. 44. This shews, that prime numbers cannot be divided by other numbers; and, on the other hand, that the simple factors of compound numbers are found most conveniently, and with the greatest certainty, by seeking the simple, or prime numbers, by which those compound numbers are divisible. But for this *Division* is necessary; we shall therefore explain the rules of that operation in the following chapter. * There is a Table at the end of a German book of arithmetic, published at Leipsic, by Poetius, in $1728$, in which all the numbers from $1$ to $10000$ are represented in this manner by their simple factors. F. T.