Elementary arithmetic, with brief notices of its history... by Robert Potts.

2
Every prime number is prime to all numbers which are not multiples of itself.
Two or more numbers are said to be coommznsurable when any
integral number will measure or divide each of them exactly; and if
no integral number can be found which will exactly measure each of
them, they are said to be incomnensurable. Unity being a measure of
every number, is not considered as a measure.
DEF. A number which is produced by the multiplication of two or
more numbers is called a common multiple of each of them; as 12 is 3
times 4, and 4 times 3; 12 is a multiple of 3 and a multiple of 4, and
therefore a common multiple of 3 and 4.
DEF. The least commnon nultiple of two or more numbers is that
number which is the least multiple of each of the given numbers.
Thus 12 is the least common multiple of 4 and 6; for 12 is 3 times 4,
and 2 times 6; and there is no number less than 12 which contains
4 and 6 a less number of times.
Or. A number which is divisible by two or more numbers is also
defined to be a common multiple of these numbers; and the least
number which is so divisible is their least common multiple. Thus
12 is a common multiple of 2 and 3; for 12 is exactly divisible by
2 and by 3: but 12 is the least common multiple of 4 and 6.
If one number measure another, it will measure any multiple of
that number; as if 3 measure 6, it will measure 2, 3, 4, 5, &c.,
times 6.
If one number be a common measure of two other numbers, it will
measure their sum and their difference; as if 3 measure 12 and 18, it
will also measure the sum 30 and the difference 6.
If one number measure two other numbers, it will measure the
sum and difference of any multiples of them; as if 3 measure 12 and
18, it will measure the sum and difference of any multiples of 12 and
18, as 4 times 12 and 5 times 18; that is, 3 will measure 138 and 42.
2. PaoP.-To find when a number is divisible by 9, and by 11, numbers
which are one less and one greater than the scale of notation.
First, let the number 7236 be taken, and divided into parts which
are, and which are not divisible by 9.
Then 7236 = 7000 + 200 + 30 + 6
and 7000=7 x 1000=7 x 999 + 7
200 =2 x 100=2 x 99 + 2
30=3 x 10=3 x 9 + 3
6= 6.. 7236 = (7x999 + 2 x 99 + 3 x 9) + (7 + 2 + 3 + 6)
The number 7236 being divided into two parts, the first of which
is obviously divisible by 9; and the second being the sum of the digits,
is also divisible by 9; it follows that the whole number 7236 is
divisible by 9 if the sum of its digits be divisible by 9. In the same