A student asks:

I don’t get the Venn diagram method for highest common factor and least common multiple. Do you have any other suggestions?

As it happens, I do. I’m assuming you’re OK with finding the prime factorisation of a number using (for example) a factor tree. In this example, I’m going to find the HCF and LCM of 84 and 70.

First, I work out the prime factors of each: $84 = 2^2 \times 3 \times 7$ and $70 = 2 \times 5 \times 7$, and write them out like this:

84 $2^2$ $3^1$ $5^0$ $7^1$
70 $2^1$ $3^0$ $5^1$ $7^1$

Notice that each factor is in its own column, and when a factor is in one list but not the other, I’ve added it in with a 0th power – this doesn’t change anything, because $a^0 = 1$, for all sensible $a$.

Now, to find the highest common factor, you take the LOWEST power from each column and multiply the numbers together. In this case, that would be $2^1 \times 3^0 \times 5^0 \times 7^1 = 2 \times 7 = 14$. And, if you check, 14 is a common factor to 70 ($=5\times 14$) and 84 ($=6\times 14$). Since 5 and 6 have no non-trivial common factors, it’s the highest common factor.

To find the least common multiple, you do the opposite: you find the HIGHEST power in each column and multiply the numbers together. Here, it’s $2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420$. Again, it’s worth checking: $420 = 5 \times 84$ and $420 = 6 \times 70$.

Interesting, don’t you think, that 5 and 6 crop up in both of the checks. You might think there was a reason for that, mightn’t you?

* Edited 2015-10-05 to correct a mistyped number. Thanks, Mark!