I usually do it by approximation most of the time. I estimate and then I multiply to see how close I got.
Multiplication is easier. 720 / 13 for this example and 1.3% looks pretty easy this way IMO, you just need a few iterations. I immediately know the result of using 10, so I'm already down to 720/130, or 72/13. Which shows me I did something stupid because I could have started that way and not forgotten what "%" means in the first place... Anyway, using 5 as a very lazy first estimate gives me 5 * 13=65 - certainly manageable in most heads. Trying 6 gives me 78.
Looking at both results, which have about the same distance from the target value, next I'll try 5.5 and that's 5 * 13 + a half of 13, still easy, 71.5. Close enough. Multiply by ten to compensate for the initial simplification, I get 55. That's really close to the result (~53.7, using the rule of 72 it is ~55.4).
I try to segment such calculations into simpler ones, instead of doing one algorithm, and finding shortcuts. For the 5.5 I use the shortcut that it's half of ten which always is easy, and that I can split it into two easy parts. To me it's more fun that way, even if I could also do the algorithm. It's supposed to be fun and being lazy is my goal.
For division you can often find shortcuts as well. Dividing by two, and powers of two subsequently, is simple enough. Any number you can divide by makes the number you have left easier. If you start from two you get prime factorization. You just have to remember a few numbers to put them together again (i.e. multiply) afterwards. Or any number less than ten really. Here too I would try to decompose. To me merely applying the algorithm is just work and maybe counts as mental exercise, but the decomposition and finding of little shortcuts is more fun and an exercise too.