Last digit! -2

$2137^{753}\equiv 7^{753}\pmod{10}\\ 7^{\lambda(10)}\equiv 7^{4}\pmod{10}\\ \implies \boxed{753\pmod{4}}\\ 753\equiv 1\pmod{4}\\ \implies 7^{753}\equiv 7^{1}\equiv \boxed{7}\pmod{10}$ Carmichael for the win!