Elements of Algebra

16 ELEMENTS SECT. I. CHAP. VI. *Of the Properties of Integers, with respect to their Divisors.* 58. As we have seen that some numbers are divisible by certain divisors, while others are not; it will be proper, in order to obtain a more particular knowledge of numbers, that this difference should be carefully observed, both by distinguishing the numbers that are divisible by divisors from those which are not, and by considering the remainder that is left in the division of the latter. For this purpose, let us examine the divisors; $2, 3, 4, 5, 6, 7, 8, 9, 10, \&c.$ 59. First let the divisor be $2$; the numbers divisible by it are, $2, 4, 6, 8, 10, 12, 14, 16, 18, 20, \&c.$ which, it appears, increase always by two. These numbers, as far as they can be continued, are called *even numbers*. But there are other numbers, viz. $1, 3, 5, 7, 9, 11, 13, 15, 17, 19, \&c.$ which are uniformly less or greater than the former by unity, and which cannot be divided by $2$, without the remainder $1$; these are called *odd numbers*. The even numbers may all be comprehended in the general expression $2a$; for they are all obtained by successively substituting for $a$ the integers $1, 2, 3, 4, 5, 6, 7, \&c.$ and hence it follows that the odd numbers are all comprehended in the expression $2a + 1$, because $2a + 1$ is greater by unity than the even number $2a$. 60. In the second place, let the number $3$ be the divisor; the numbers divisible by it are, $3, 6, 9, 12, 15, 18, 21, 24, 27, 30$, and so on; which numbers may be represented by the expression $3a$; for $3a$, divided by $3$, gives the quotient $a$ without a remainder. All other numbers which we would divide by $3$, will give $1$ or $2$ for a remainder, and are consequently of two kinds. Those which after the division leave the remainder $1$, are, $1, 4, 7, 10, 13, 16, 19, \&c.$ and are contained in the expression $3a + 1$; but the other kind, where the numbers give the remainder $2$, are, $2, 5, 8, 11, 14, 17, 20, \&c.$ which may be generally represented by $3a + 2$; so that all numbers may be expressed either by $3a$, or by $3a + 1$, or by $3a + 2$.