Can someone check if my work is correct on this problem?

Given $\displaystyle 1728=2^63^3$. Show that if $\displaystyle (a,7)=1, (a,13)=1, (a,19)=1$ then $\displaystyle a^{1728}\equiv 1 mod(7); a^{1728}\equiv 1 mod (13); a^{1728}\equiv 1 mod (19)$

By FLT, we have $\displaystyle a^6\equiv 1 mod(7)$,

$\displaystyle a^{1728}\equiv (a^6)^{288}\equiv 1^{288}\equiv 1 mod (7)$

We can prove the other two similarly.

Then the follow question is to show that $\displaystyle a^{1728}\equiv 1 mod (1729)$ using the result above and given that $\displaystyle 1729=7\times13\times19$. Can someone help me make the argument here?