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

My first intuition is to use modular arithmetic. 
Note that 3986 = 4000 - 14 and so 3987 = 3 mod 4.  Now
$$(3987)^2 \space mod \space 4 = 9 \space mod \space 4 = 1$$
$$\text{and so }3987^{12} = 1 \space mod \space 4$$
Now, we also use the fact that
$$4365 \space mod \space 4 = 1$$
$$\text{and so }4365^12 = 1 \space mod \space 4$$
So, one side is 2 mod 4, but the other side is a square, since
$$4472^{12} = (4472^6)^2$$
Now, if 4472 is 2 mod 4, then
$$4472^2 = 0 \space mod \space 4$$
$$\text{and so }4472^{12} = 0 \space mod \space 4$$
Hence, this proves that the sides are not equivalent mod 4 and we are done. 
