Remainder again....:-(:-(

Euler number E(11) = 11(1-1/11) = 10

Now, if D^m is the number whose remainder is to be found out.then D^E gives remainder 1 when divided by N

here, D= 50 m = 51^52

We should express m in terms of E. So 51^53 is to be written in the form of 10k + a. Now unit digit of 51^53 = 1 => 51^53 = 10k + 1.

=> 50^51^53 = 50^(10k + 1) = 50mod11 = 6mod11.

thus, Answer = 6 .

Since $\gcd(50, 11)=1$ , we can use Euler's Theorem. First, we need to find $53$ modulo $\phi(10)=4$ , second, we need to find ${51}^{53}$ modulo $\phi(11)=10$ , and the last we need to find $\large{{50}^{{51}^{53}}}$ modulo $11$ .

$53\equiv1\pmod{4}\\{51}^{53}\equiv{51}^{1}\equiv1\pmod{10}\\\large{{50}^{{51}^{53}}}\equiv{50}^{1}\equiv6\pmod{11}$