Conventional Approach

Using the technique of last two digits:

The question is to find the last two digits of $(1073{)}^{71}$ (which is nothing but the remainder when the number is divided by 100).

We will solve this using a technique as explained below and also in our demo tutorial in arithmetic.

$(1073{)}^{71}=(..73{)}^{71}=(..73{)}^{68}\times (..73{)}^3$
$=(.{.73}^2\times .{.73}^2{)}^{17}\times (..73{)}^3$
$=(..29\times ..29{)}^{17}\times (..73{)}^3$
$=(..41{)}^{17}\times (..73{)}^3$
$=81\times 29\times 73=...77$

Shortcut of a shortcut!
Using answer options:

From the power cycle of 3 (3,9,7,1) -

71= 4k+3

Thus, the last digit should be 7. The only answer option ending with 7 is option (b).