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

6
last quotient is unity, the divisors will be the prime factors, which,
when multiplied together, will reproduce the given composite number.
It is obvious that any composite number can only be resolved into
one set of prime factors.
If two numbers be resolved into their prime factors, it will at once
appear on inspection what factors are common to the two numbers;
the product of these factors will be the greatest common measure of
the two numbers.
Thus may le found the greatest common measure of 24 and 36:Here, 24=2 x 2 x 2 x 3, and 36-2 x 2 x 3 x 3,
The factors, 2, 2, 3, are common;
And, therefore, 2 x 2 x 3 = 12, is the greatest common measure of
24 and 36.
The same method can be applied to find the least common multiple
of two numbers.
In a similar way, if two numbers be resolved into their prime
factors, by inspection may be seen what factors are common, and what
factors are not common to the two numbers. The product of these
factors will obviously be the least multiple of each of the two numbers,
and consequently their least common multiple.
Thus may be found the least common multiple of 24 and 36:
Here, 24 - 2 x 2 x 2 x 3 and 36 - 2 x 2 x 3 x 3.
The prime factors 2, 2, 3, are common, and 2, 3 are not common.
Hence the product 2 x 2 x 3 x 2 x 3, or 72, of the prime factors
2x2x3, which are common, and of 2 and of 3, which are not
common, is the least common multiple of 24 and 36.
The resolution of numbers into their prime factors appears to be
the more simple mode of discovering the greatest common measure
and the least common multiple of two or more numbers. When the
numbers are small this method can be applied with advantage. But
when the given numbers are large there will arise some difficulty in
discovering by inspection when a number is exactly divisible and
when not divisible by a large prime number. This difficulty is
obviated by a process by which can be determined with exactness the
greatest common measure and the least common multiple of any
numbers how great soever they may be.
6. PROP. To find the greatest common mneasure, or the greatest con)mmon
divisor of any two numbers. (Euc. vii. 2.)
Let it be required to find the greatest common divisor of 189
and 224. 189)224(1
Here, 224 divided by 189 gives quot. 1, and rem. 35 189
35)189(5
189,, 35,, 5,, 14 175
14)35(2
35, 14,, 2, 7 28
14,,?7,, 2 0, 7)14(2
1 4 7 0o 2 i 14
The lan-t divisor, 7, is a common measure of 189 and 224.