prime factorize numbers in my head. i used to just pick random numbers i saw around esp. at math competitions. picked up some useful mental math tricks this way
the main mental math trick is to check divisibility by repeatedly subtracting large multiples of the number. like check divisibility by 7 by repeatedly subtracting e.g. 70, 1400, 4900, etc. also the reverse, like subtracting into (or allowing negatives)
there's a trick based on digits like the 9 trick but harder but it's pretty annoying and doesn't generalize well. subtracting multiples works fine up to 2-digit primes