ALGEBRA problem

We have to make a really important observation to solve this problem. Note that

$1,005,010,010,005,001 = \\ 1\times 10^{15} + 5 \times 10^{12} + 10\times 10^9 + 10\times 10^6 + 5 \times 10^3 + 1\times 10^0\\ $

This value is equal to

$(10^3+1)^5$, which is equal to $1001^5$

Now, we have to find the largest prime factor of 1001. 

We know that $1001=13*11*7$

This means the largest prime factor is 13. 

Thanks! :)