Elements of Algebra

CHAP. VI. OF ALGEBRA. 17 61. Let us now suppose that $4$ is the divisor under consideration; then the numbers which it divides are, $4, 8, 12, 16, 20, 24, \&c.$ which increase uniformly by $4$, and are comprehended in the expression $4a$. All other numbers, that is, those which are not divisible by $4$, may either leave the remainder $1$, or be greater than the former by $1$; as, $1, 5, 9, 13, 17, 21, 25, \&c.$ and consequently may be comprehended in the expression $4a + 1$: or they may give the remainder $2$; as, $2, 6, 10, 14, 18, 22, 26, \&c.$ and be expressed by $4a + 2$; or, lastly, they may give the remainder $3$; as, $3, 7, 11, 15, 19, 23, 27, \&c.$ and may then be represented by the expression $4a + 3$. All possible integer numbers are contained therefore in one or other of these four expressions: $4a, 4a + 1, 4a + 2, 4a + 3$. 62. It is also nearly the same when the divisor is $5$; for all numbers which can be divided by it are comprehended in the expression $5a$, and those which cannot be divided by $5$, are reducible to one of the following expressions: $5a + 1, 5a + 2, 5a + 3, 5a + 4$; and in the same manner we may continue, and consider any greater divisor. 63. It is here proper to recollect what has been already said on the resolution of numbers into their simple factors; for every number, among the factors of which is found $2$, or $3$, or $4$, or $5$, or $7$, or any other number, will be divisible by those numbers. For example; $60$ being equal to $2 \times 2 \times 3 \times 5$, it is evident that $60$ is divisible by $2$, and by $3$, and by $5$*. * There are some numbers which it is easy to perceive whether they are divisors of a given number or not. 1. A given number is divisible by $2$, if the last digit is even; it is divisible by $4$, if the two last digits are divisible by $4$; it is divisible by $8$, if the three last digits are divisible by $8$; and, in general, it is divisible by $2^n$, if the $n$ last digits are divisible by $2^n$. 2. A number is divisible by $3$, if the sum of the digits is divisible by $3$; it may be divided by $6$, if, beside this, the last digit is even; it is divisible by $9$, if the sum of the digits may be divided by $9$. 3. Every number that has the last digit $0$ or $5$, is divisible by $5$. C