Definition of “common multiple”

[xilman]

Originally Posted by drkirkby

I note that maxima gives lcm(6,4) as 12 and not 24.

The clue is in the word "common".

12 is a multiple of 4, being 3*4
12 is a multiple of 6, being 2*6

So 12 is a multiple of both 4 and 6 --- it has those factors in common with both of them.

[charybdis]

Originally Posted by drkirkby

So when computing the common factors of m and n, you have to factorise m and n if they are not prime? The OU doesn’t say that.

As a1call has mentioned, gcd(m,n) can be found very quickly without factorizing m and n, by using the Euclidean algorithm. Then use lcm(m,n) = mn/gcd(m,n).

[Dr Sardonicus]

Originally Posted by drkirkby

<snip>
Common multiple is not a term that I have come across before, although I have come across least common multiple (lcm).
<snip>

This statement is self-contradictory. The term "common multiple" is included in "least common multiple."

Discerning the meaning of "common multiple" is merely a matter of knowing two basic definitions. Here, common clearly means "shared." And a multiple of an integer is that integer multiplied by an integer.

Originally Posted by drkirkby

But the OU gets a smaller number (12) than multiplying 6 and 4.

Is 12 a multiple of 6? Yes, 12 = 6*2. Is 12 a multiple of 4? Yes, 12 = 4*3. So, 12 is a multiple of both 4 and 6; that is, it is a common multiple of 4 and 6.

[drkirkby]

Originally Posted by drkirkby

So when computing the common factors of m and n, you have to factorise m and n if they are not prime? The OU doesn’t say that. Given that this is a very basic book on maths, not mentioning that seems wrong.

Having read this OU book more, it states that there’s an efficient method, known as Euclid’s algorithm, for finding the lowest common multiple and highest common factors without factorising. So my earlier comment was unfair to the OU. Their material is very good quality.