180
I’ll explain it as I remember it and use it. The key is that 9 = 0. You add the digits together and you keep reducing until you get to less than 9.
Take this addition problem:
371
+ 216
840
____
1427
Is the answer right or wrong? To check it, you cast out nines.
3 + 7 + 1 = 11 and keep going: 1 + 1 = 2
2 + 1 + 6 = 9; 9 = 0
8 + 4 + 0 = 12; 1 + 2 = 3
Now add those: 2 + 0 + 3 = 5
AND the same with the sum:
1 + 4 + 2 + 7 = 14; 1 + 4 = 5
So now you know your addition was right. If it weren’t, the answers wouldn’t match.
(To speed the check, cast out 9’s as you go: 371 -> 3 + 7 = 10, and 10 = 1; and then 1 + 1 = 2)
You can do this with any arithmetic operation, addition, subtraction, multiplication, or division, and the answer will match the terms in this same way.
If your computation is off by 9 or a multiple of 9, the check won’t catch it, but otherwise, yes.
Also I remember a bookkkeeper’s telling me that if you’re off by 9, it’s probably a transposition error.
So now, @Brian1946, who might have been in seventh grade at the same time that I was, maybe this points to the explanation you’re looking for; because of course 3 squared is 9.