2018 Regional Set $2$ Higher Secondary $P6$

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samiul_samin
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Joined:Sat Dec 09, 2017 1:32 pm
2018 Regional Set $2$ Higher Secondary $P6$

Unread post by samiul_samin » Thu Jan 10, 2019 11:09 am

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

NABILA
Posts:35
Joined:Sat Dec 15, 2018 5:19 pm
Location:Munshigonj, Dhaka

Re: 2018 Regional Set $2$ Higher Secondary $P6$

Unread post by NABILA » Wed Jan 16, 2019 3:16 pm

Didn't understood. Please give some hints.
Wãlkîñg, lõvǐñg, $mīlïñg @nd lìvíñg thě Lîfè

Toky
Posts:3
Joined:Mon Jan 14, 2019 11:36 pm

Re: 2018 Regional Set $2$ Higher Secondary $P6$

Unread post by Toky » Thu Jan 17, 2019 12:49 pm

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

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 $1000000$.

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