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What are the last seven digits of the binary form of
$65^{2016}-65^{2015}$ ?

Didn't understood. Please give some hints.

$65^{2016} - 65^{2015} = 65^{2015}(65-1) = 65^{2015}\times64$

Now let's convert $64$ into Binary. $(64)_2 = 1000000$.
Notice that there are $6$ $zero$s at the end of the number. That means if we multiply any number with that number, we will get $6$ $zero$s at end.

Let's determine the $7^{th}$ number from the end.
Binary of $65$ is $1000001$. The last digit is $1$. That means the last digit of any power of this number is $1$.

So, the last $7$ digits number of $65^{2016} - 65^{2015}$ are $1000000$.