The number $555,555$ can decompose, as the product of two factors of three digits, in how many ways? 
The number $555,555$ can decompose, as the product of two factors of three digits, in how many ways? 

I've seen the answer to the question, and there is only one way: 
Since $555, 555 = 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 37$, the only way to combine the factors to achieve expressing it as a product of two three-digit numbers is $(3 \cdot 7 \cdot 37) (5 \cdot 11 \cdot 13)$. Regardless of this, I struggle to understand how the answer was formulated. Can someone show me the procedure?
Sorry if the question is poorly phrased, it is a rough translation of the original problem in Spanish. 

I think the procedure might be based on Fermat method.
We can calculate that the number $555555$ can be expressed as a difference of two squares:
$746^2 - 31^2=556516-961$
Number $31$ is the distance from $746$ both ways which gives $715$ and $777$ accordingly. From here the smallest prime factor of:
$777$ is number $3$ which equals to $3\cdot259$ and for
$715$ is number $5$ which equals to $5\cdot143$
Further factorisation results in final solution where $143=11\cdot13$ and $259=7\cdot37$ therefore:
$(5\cdot11\cdot13)(3\cdot7\cdot37)$ can satisfy equation. 
