If n is an integer, how many values of n will give an integral value of $51{\mathrm{n}}^2+17\mathrm{n}+6$?

The sum of the last $10$ digits of the sum of the expression:

$\left(1^1\times 2^2\times 3^3\times 4^4\times 5^5\right)+\left(1^6\times 2^7\times 3^8\times 4^9\times 5^{10}\right)+\left(1^{11}\times 2^{12}\times 3^{13}\times 4^{14}\times 5^{15}\right)+\dots +\left(1^{96}\times 2^{97}\times 3^{98}\times 4^{99}\times 5^{100}\right)$ is:

The sum of:

$\left(2^2+4^2+6^2+\dots +{100}^2\right)-\left(1^2+3^2+5^2+\dots +{99}^2\right)$ is: