Suppose I could write every numbers as a sum of $3$ cubes in (mod $a_1$) and I could write every number as a sum of 3 cubes (mod $a_2$), where $a_1$ and $a_2$ are coprime. Can we show that every number in (mod $a_1a_2$) can be written as a sum of $3$ cubes?
It seems to me that this will be an application of CRT somehow but I could not quite see how to apply CRT here. I have thought about taking every element in (mod $a_1a_2$) and project onto (mod $a_1)$ and (mod $a_2$) separately. In particular, every element after the projection can be written as a sum of $3$ cubes but I am not sure how to proceed further.
Many thanks!
CRT works. If $n\equiv b^3+c^3+d^3\bmod{a_1}$ and $n\equiv r^3+s^3+t^3\bmod{a_2}$ and $\gcd(a_1,a_2)=1$ then you can find $u,v,w$ such that $u\equiv b\bmod{a_1}$, $u\equiv r\bmod{a_2}$ and similarly for $v,c,s$ and $w,d,t$. Then you'll have $u^3+v^3+w^3\equiv n\bmod{a_1a_2}$.