**J Alan Smith considers this misleading metaphor**

**R**eaders of a certain age will no doubt recall terms from Mathematics like the Highest Common Factor (HCF), the Lowest Common Multiple (LCM), and, a variation of the latter, the Lowest Common Denominator (LCD), used when adding fractions. Perhaps fewer can remember the definitions of the terms. This article provides the necessary definitions. The intention is not to provide a revision course in Mathematics but to test whether the metaphorical use of LCM and LCD is validly based in Mathematics or whether it is simply a hackneyed practice that has no logical basis.

Each of the terms HCF and LCM applies to a finite set of positive whole numbers. A finite set of whole numbers necessarily has a finite set of common factors for it could have no common factor larger than the smallest number in the set: the HCF is the largest of these. In contrast, a finite set of whole numbers has an infinite set of common multiples for, given any common multiple, other common multiples could be obtained by multiplying that common multiple by 2, 3, 4, …: the LCM is the smallest of the common multiples. The HCF is never greater than the LCM and, except when all the numbers in the set are equal, the HCF is strictly less that the LCM. Looking at them in another way: the HCF is a factor of every number in the set; every number is the set is a factor of the LCM.

For an example let us take the following set of numbers: 8, 12, and 20. Express each number as the product of its prime factors. If a prime is a factor of some but not all of the numbers, where it is not present insert that prime to the power of 0, for any number raised to the power of 0 is equal to 1. Further, where a prime is a factor to the power of 1, I have inserted the 1, which is usually taken as implied, in order to demonstrate the process more easily.

8 = 23 x 30 x 50

12 = 22 x 31 x 50

20 = 22 x 30 x 51

The HCF is the product of each prime to the *lowest* power at which it occurs in the factors of any of the numbers: 22 x 30 x 50 i.e. 4. The LCM is the product of each prime to the *highest* power at which it occurs in the factors of any of the numbers: 23 x 31 x 51 i.e. 120.

Using terms from Mathematics or the physical sciences as a metaphor is acceptable as long as one respects the logical structure of the original. In the case of a finite set of numbers and its HCF and its LCM, we could legitimately extend its use to a finite set of objects with certain attributes: its HCF could be the object that has only those attributes common to all objects in the set and its LCM is the object that has all the attributes of every object in the set.

Let us take an example from *Gaudy Night* by Dorothy L. Sayers:

*Sunday evening prayers. The College was undenominational, but some form of Christian worship was held to be essential to community life. The chapel, with its stained-glass windows, plain oak panelling and unadorned Communion Table was a kind of Lowest Common Multiple of all sects and creeds.*

It is clear that what Miss Sayers is describing is not the Lowest Common Multiple but the Highest Common Factor. I am not seeking to denigrate Miss Sayers: it is a tribute to her prose that she has clearly illustrated the underlying fallacy. Unfortunately, the metaphorical misuse of LCM and LCD is and has been so common that individual practitioners are absolved from responsibility, taking the meaning of these terms from the way they are misused. Should one worry about so well-established an error? I can only state that:

* * *our concern was speech, and speech impelled us*

* * *to purify the dialect of the tribe.*