Elements of Algebra

CHAP. VI. OF ALGEBRA. 19 three by three, four by four, &c. till we arrive at the number proposed. 66. It must here be particularly observed, that every number is divisible by $1$; and also, that every number is divisible by itself; so that every number has at least two factors, or divisors, the number itself, and unity: but every number which has no other divisor than these two, belongs to the class of numbers, which we have before called *simple*, or *prime numbers*. Except these simple numbers, all other numbers have, beside unity and themselves, other divisors, as may be seen from the following Table, in which are placed under each number all its divisors *. TABLE. $1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6 \quad 7 \quad 8 \quad 9 \quad 10 \quad 11 \quad 12 \quad 13 \quad 14 \quad 15 \quad 16 \quad 17 \quad 18 \quad 19 \quad 20$ $1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1$ $\quad 2 \quad 3 \quad 2 \quad 5 \quad 2 \quad 7 \quad 2 \quad 3 \quad 2 \quad 11 \quad 2 \quad 13 \quad 2 \quad 3 \quad 2 \quad 17 \quad 2 \quad 19 \quad 2$ $\quad \quad \quad 4 \quad \quad 3 \quad \quad 4 \quad 9 \quad 5 \quad \quad 3 \quad \quad 7 \quad 5 \quad 4 \quad \quad 3 \quad \quad 4$ $\quad \quad \quad \quad \quad 6 \quad \quad 8 \quad \quad 10 \quad \quad 4 \quad \quad 14 \quad 15 \quad 8 \quad \quad 6 \quad \quad 5$ $\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 6 \quad \quad \quad \quad 16 \quad \quad 9 \quad \quad 10$ $\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 12 \quad \quad \quad \quad \quad \quad 18 \quad \quad 20$ $1 \quad 2 \quad 2 \quad 3 \quad 2 \quad 4 \quad 2 \quad 4 \quad 3 \quad 4 \quad 2 \quad 6 \quad 2 \quad 4 \quad 4 \quad 5 \quad 2 \quad 6 \quad 2 \quad 6$ $\quad$ P. P. $\quad$ P. $\quad$ P. $\quad \quad \quad$ P. $\quad$ P. $\quad \quad \quad$ P. $\quad \quad$ P. 67. Lastly, it ought to be observed that $0$, or *nothing*, may be considered as a number which has the property of being divisible by all possible numbers; because by whatever number $a$ we divide $0$, the quotient is always $0$; for it must be remarked, that the multiplication of any number by *nothing* produces *nothing*, and therefore $0$ times $a$, or $0a$, is $0$. * A similar Table for all the divisors of the natural numbers, from $1$ to $10000$, was published at Leyden, in $1767$, by M. Henry Anjema. We have likewise another table of divisors, which goes as far as $100000$, but it gives only the least divisor of each number. It is to be found in Harris's Lexicon Technicum, the Encyclopédie, and in M. Lambert's Recueil, which we have quoted in the note to p. $11$. In this last work, it is continued as far as $102000$. F. T. c 2