A problem from number theory

For discussing Olympiad Level Number Theory problems
Naeem Mashkur
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A problem from number theory

Unread post by Naeem Mashkur » Tue Jun 15, 2021 7:03 am

What is the maximum power of $2$ which divides $10^{1002} – 4^{501} ?$
Last edited by tanmoy on Tue Jun 15, 2021 8:41 pm, edited 1 time in total.
Reason: Latexed

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Mehrab4226
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Location:Dhaka, Bangladesh

Re: A problem from number theory

Unread post by Mehrab4226 » Tue Jun 15, 2021 10:30 pm

Naeem Mashkur wrote:
Tue Jun 15, 2021 7:03 am
What is the maximum power of $2$ which divides $10^{1002} – 4^{501} ?$
$10^{1002}-4^{501} = 2^{1002}5^{1002}-2^{1002}=2^{1002}(5^{1002}-1) = 2^{1002}(5^{501}+1)(5^{501}-1)$
Now in mod $4$,
$5^{501}+1 \equiv 2$
So it is a multipe of $2$ but not $4$.
In mod $8$ we get,
$5^{501}-1 \equiv 5\times (5^2)^{250}-1 \equiv 5-1 \equiv 4$ $[5^2 \equiv 1]$. So it is not divisible by $8$.
So the the the answer is $\boxed{2^{1005}}$
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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