Mistaken counterexample to FLT; where's the mistake?
Solution 1:
7 divides both 91 and 56, but not 121.
Solution 2:
The child could have plugged $121^2 - 56^2 - 91^2$ into a calculator and seen that it's greater than 0, so there's no way in hell it'd work for $x > 2$.
Or he could have realized there was something off about the last digits of the numbers. What's so special about $1$ and $6$?
EDIT: You don't even need any clever tricks for this one, just a good sense of estimation. $121^2 > 120^2 = 14400$. $56^2 < 60^2 = 3600$. $91^2 < 100^2 = 10000$. Since $10000+3600 < 14400$, then $91^2 + 56^2 < 121^2$.