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

224 =189 x 1+ 35, 189 = 35 x 5+14,
35=14 x 2+ 7, 14=7 x 2+0.
Iiere 7, the last divisor, measures 14, by the units in 2.
Next, 7 measures 14 x 2, a multiple of 14, and 14 x 2 + 7, or 35.
Thirdly, 7 also measures 35 x 5, and 35 x 5 +14, or 189.
Fourthly, since 7 measures 189 and 85, its measure 189 x 1 + 35,
or 224.
Therefore 7 is a common neasure of 189 and 224.
And there is no number greater than 7 which will measure 189
and 227.
224-189 x 1=35, 189-35 x 5 =14, 35-14 x2=7
For, if possible, let some number greater than 7 measure 189
and 224.
Then this number measures 224-189 = 35, their difference.
It also measures 35 x 5, and 189-35 x 5 or 14.
And, thirdly, it measures 35-14 x 2 or 7.
That is, a number greater than 7 measures 7; which is impossible.
Wherefore 7, the last divisor, is the greatest common measure of
189 and 224.
Hence, if the greater of two numbers be divided by the less, and
the preceding divisor always by the last remainder, until there be no
remainder, the last divisor is the greatest common measure of the two
numbers.
If the last divisor be unity, the two numbers are prime to each
other, and have no common measure.
To find the greatest common measure of more numbers than two:1. Of three numbers.
-Iaving found the G.C.M. of two of the three numbers, the G.C.M.
of this number and the third number will be the G.C.M. of the three
numbers.
2. Of four numbers.
Having found the G.C.M. of three of the numbers, the G.C.M. of
this number and the fourth number will be the G.C.M. of the four
numbers.
And in a similar way for five, six, seven, &c., numbers.
7. Pr:or. —b find the least common multiple of tco numbers.
Let 18 and 24 be two numbers, of which the greatest common
measure is 6.
Then 18 -6 x 3, and 24 = 6 x 4, also 18 x 24 = 6 x 3 x 6 x 4,
And obviously the least common multiple of the two numbers will
consist of the product of all the prime factors in the two numbers;
or the least common multiple of 18 and 24 = 6 x 4 x 3 or 18 X4 = 72.
And there is no integral number less than 72, which is a less
multiple of 18 and 24;