Elements of Algebra

10 ELEMENTS SECT. I. should only multiply $36$ first by $3$, and then the product $108$ by $4$, in order to have the whole product of the multiplication of $12$ by $36$, which is consequently $432$. 36. But if we wished to multiply $5ab$ by $3cd$, we might write $3cd \times 5ab$. However, as in the present instance the order of the numbers to be multiplied is indifferent, it will be better, as is also the custom, to place the common numbers before the letters, and to express the product thus: $5 \times 3abcd$, or $15abcd$; since $5$ times $3$ is $15$. So if we had to multiply $12pqr$ by $7xy$, we should obtain $12 \times 7pqrxy$, or $84pqrxy$. CHAP. IV. *Of the Nature of whole Numbers, or Integers, with respect to their Factors.* 37. We have observed that a product is generated by the multiplication of two or more numbers together, and that these numbers are called *factors*. Thus, the numbers $a, b, c, d$, are the factors of the product $abcd$. 38. If, therefore, we consider all whole numbers as products of two or more numbers multiplied together, we shall soon find that some of them cannot result from such a multiplication, and consequently have not any factors; while others may be the products of two or more numbers multiplied together, and may consequently have two or more factors. Thus $4$ is produced by $2 \times 2$; $6$ by $2 \times 3$; $8$ by $2 \times 2 \times 2$; $27$ by $3 \times 3 \times 3$; and $10$ by $2 \times 5$, &c. 39. But on the other hand, the numbers $2, 3, 5, 7, 11, 13, 17$, &c. cannot be represented in the same manner by factors, unless for that purpose we make use of unity, and represent $2$, for instance, by $1 \times 2$. But the numbers which are multiplied by $1$ remaining the same, it is not proper to reckon unity as a factor. All numbers, therefore, such as $2, 3, 5, 7, 11, 13, 17$, &c. which cannot be represented by factors, are called *simple*, or *prime numbers*; whereas others, as $4, 6, 8, 9, 10, 12, 14, 15, 16, 18$, &c. which may be represented by factors, are called *composite numbers*. 40. Simple or *prime numbers* deserve therefore particular attention, since they do not result from the multiplication of