Modular arithmetic How do I prove the following inequality with modular arithmetic? (No use of Fermat's last theorem is allowed.)
$$3987^{12} + 4365^{12} \neq 4472^{12}$$

Note that $3|3987,3|4365,3\not| 4472$ so we have $3^{12}a^{12}+3^{12}b^{12}=4472^{12}\implies 3|4472,\text{ and }3\not| 4472\implies 3987^{12}+4365^{12}\ne 4472^{12}.$
