jmc

We notice that $450=221+229,$ so that must be the connection. The Pythagorean Theorem tells us $$60^2+221^2=229^2$$ so $$229^2-221^2=60^2.$$ The difference of squares factorization tells us $$(229-221)(229+221)=3600$$ and taken modulo 3599 we get $$8\cdot 450\equiv 1(\mathrm{mod}3599).$$ The answer is $\boxed{8}$.