I have a student (age 14) who is valiantly striving to improve his algebra. He knows the "rules", and he more-or-less manages to apply them correctly, and hence is generally able to tackle the "simplify this" which his curriculum routinely throws at him.
But then he encounters something like:
$$-14 x + 21 = 7 x^2$$
and he completely fails to notice that he can divide every term by $7$.
When I pointed out to him that cancelling big numbers out of his expressions (and in this context, "big" seems to be anything greater than or equal to $2$) his reaction was: "How can I tell what number to cancel by?"
And at that point it occurred to me that was unable to recognise that, in the expression above, he could not tell that $7$ was a divisor of each of $14$ and $21$.
Back in my day, we learned our "times tables", and hence recognising by sight every $2$-digit number's prime factors is instant, so I never even thought that would be a problem.
Now I realise that because pupils are no longer expected to learn arithmetic, they are also unable to "recognise" numbers in this way.
Apart from giving this dude lots of practice doing basic arithmetic (which doesn't work because he doesn't do it), what strategies can I suggest for him so he can gain facility at "cancelling down" composite numbers like this?