What are the last seven digits of the binary form of
$65^{2016}-65^{2015}$ ?
2018 Regional Set $2$ Higher Secondary $P6$
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Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
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Re: 2018 Regional Set $2$ Higher Secondary $P6$
Didn't understood. Please give some hints.
Wãlkîñg, lõvǐñg, $mīlïñg @nd lìvíñg thě Lîfè
Re: 2018 Regional Set $2$ Higher Secondary $P6$
$65^{2016} - 65^{2015} = 65^{2015}(65-1) = 65^{2015}\times64$
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 $1000000$.
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 $1000000$.