The following are the most important properties of numbers in reference to factors:—
(i) If a number is a factor of another number, it is a factor of any multiple of that number.
(ii) If a number is a factor of two numbers, it is a factor of their sum or (if they are unequal) of their difference. (The words in brackets are inserted to avoid the difficulty, at this stage, of saying that every number is a factor of 0, though it is of course true that 0·n = 0, whatever n may be.)
(iii) A number can be resolved into prime factors in one way only, no account being taken of their relative order. Thus 12 = 2 × 2 × 3 = 2 × 3 × 2 = 3 × 2 × 2, but this is regarded as one way only. If any prime occurs more than once, it is usual to write the number of times of occurrence as an index; thus 144 = 2 × 2 × 2 × 2 × 3 × 3 = 24·32.
The number 1 is usually included amongst the primes; but, if this is done, the last paragraph requires modification, since 144 could be expressed as 1·24·32, or as 12·24·32, or as 1p·24·32, where p might be anything.
If two numbers have no factor in common (except 1) each is said to be prime to the other.
The multiples of 2 (including 1·2) are called even numbers; other numbers are odd numbers.
46. Greatest Common Divisor.—If we resolve two numbers into their prime factors, we can find their Greatest Common Divisor or Highest Common Factor (written G.C.D. or G.C.F. or H.C.F.), i.e. the greatest number which is a factor of both. Thus 144 = 24·32, and 756 = 22·33·7, and therefore the G.C.D. of 144 and 756 is 22·32 = 36. If we require the G.C.D. of two numbers, and cannot resolve them into their prime factors, we use a process described in the text-books. The process depends on (ii) of § 45, in the extended form that, if x is a factor of a and b, it is a factor of pa − qb, where p and q are any integers.
The G.C.D. of three or more numbers is found in the same way.
47. Least Common Multiple.—The Least Common Multiple, or L.C.M., of two numbers, is the least number of which they are both factors. Thus, since 144 = 24·32, and 756 = 22·33·7, the L.C.M. of 144 and 756 is 24·33·7. It is clear, from comparison with the last paragraph, that the product of the G.C.D. and the L.C.M. of two numbers is equal to the product of the numbers themselves. This gives a rule for finding the L.C.M. of two numbers. But we cannot apply it to finding the L.C.M. of three or more numbers; if we cannot resolve the numbers into their prime factors, we must find the L.C.M. of the first two, then the L.C.M. of this and the next number, and so on.