Pairs of generators in $C_{5} \times C_{5}\times C_{5}$

I want to find which pairs of elements generate $$C_{5} \times C_{5}\times C_{5}$$. I have demonstrated that each element in this group generate the group with another 50 elements, but I want to find now how they are.

For example, if I have $$a=(a, a^{2}, a^{3})\in C_{5} \times C_{5}\times C_{5}$$, I would like to understand how I have to write $$b\in C_{5} \times C_{5}\times C_{5}$$ in such a way that $$\langle a, b\rangle = C_{5} \times C_{5}\times C_{5}$$.

Any idea? thanks.

Firstly $$a=(a,a^2,a^3)$$ is nonsense, I assume you mean $$a=(g,g^2,g^3)$$ where $$g$$ generates $$C_5$$.
Fix $$a,b\in C_5\times C_5\times C_5$$. What are the orders of $$\langle a\rangle$$ and $$\langle b\rangle$$?
We can write $$\langle a,b\rangle=\{a^i,b^j|i,j\in\{0,1,2,3,4\}\}$$ (why?) so the order of $$\langle a,b\rangle$$ is...