I was noticing today that you can calculate the square of numbers by taking the number you want to square and removing the ones digit and multiplying by the number you want to square plus the ones digit. Then you square the ones digit and place it back on the number to get the square
503^2 = 253009
then tacking the 3^2 on the end gets 253009.
Tricks similar to that seem to work so far for larger numbers as well and with other 1s digits, sometimes slightly differently though. I assume it has something to do with base 10 but I am not sure on why this works or how you would write a proof of it.
Any help would be appreciated.