When I was young I was taught the multiplication table. Fine. I waited for the other shoe to drop but noone ever taught me the "division table". So I figured it out myself by doing all the long divisions by hand (this was before calculators) and I memorized all fractions from 1/1 to 12/12 .
Decades later, I was teaching a data structures and algorithms class at my old univeristy. In the class book (probably Wirth's, but I'm not sure) was a complete listing of all solutions to the 8-queens problem expressed in a compact form as 8-digit numbers. One of them jumped out at me as containing the magic sequence "428571" which are the digits that form the expansion of 3/7 = 0.428571428571.. . .
So I quickly figured out it was 255 / 7 that gave us 36.428571 as a valid solution to the 8-queens problem. I was so fond of this discovery that I made it my corporate logo when I started Syntience Inc. in 2004.