by 5, and the product multiplied by 3, is the same as 27 multiplied by 5 times 3, or 15. Also, if a number be divided by any number, and the quotient be divided by another, it is the same as if the first number had been divided by the product of the other two. For example, divide 60 by 4, which gives 15, and the quotient by 3, which gives 5. It is plain, that if each of the four fifteens of which 60 is composed be divided into three equal parts, there are twelve equal parts in all; or, a division by 4, and then by 3, is equivalent to a division by 4 × 3, or
- The following rules will be better understood by stating them in an example. If 32 be multiplied by 24 and divided by 6, the result is the same as if 32 had been multiplied by the quotient of 24 divided by 6, that is, by 4; for the sixth part of 24 being 4, the sixth part of any number repeated 24 times is that number repeated 4 times; or, multiplying by 24 and dividing by 6 is equivalent to multiplying by 4.
- Again, if 48 be multiplied by 4, and that product be divided by 24, it is the same thing as if 48 were divided at once by the quotient of 24 divided by 4, that is, by 6. For, every unit which is repeated 6 times in 48 is repeated 4 times as often, or 24 times, in 4 times 48, or the quotient of 48 and 6 is the same as the quotient of 48 × 4 and 6 × 4.
- The results of the last five articles may be algebraically expressed thus:
| ma | = | a | (85) |
|---|---|---|---|
| mb | b |
If n divide a and b without remainder,
| a/n | = | a | (86) |
|---|---|---|---|
| b/n | b | ||
| a/b | = | a | (87) |
| c | bc | ||
| ab | = a × | b | (88) |
| c | c | ||
| ac | = | a | (89) |
| b | b/c |
It must be recollected, however, that these have only been proved in the case where all the divisions are without remainder.
- When one number divides another without leaving any remainder, or is contained an exact number of times in it, it is said to be a measure of that number, or to measure