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- Feb 14, 2012

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Problem:

Prove that $27195^8-10887^8+10152^8$ is divisible by $26460$.

Attempt:

I grouped the last two terms and manipulated them algebraically and came to the point where I suspect I might have taken the wrong path...here is the last step where I stopped and don't know how to proceed.

$\dfrac{27195^8-10887^8+10152^8}{26460}=\dfrac{3^5\cdot5^7\cdot7^{14}\cdot37^8-7013(2^6\cdot3^6\cdot47^2+3^2\cdot19^2\cdot191^2)(10152^4+10887^4)}{8}$

I'd like to ask, do you think this problem can be solved using only elementary methods?

Thanks in advance.

Prove that $27195^8-10887^8+10152^8$ is divisible by $26460$.

Attempt:

I grouped the last two terms and manipulated them algebraically and came to the point where I suspect I might have taken the wrong path...here is the last step where I stopped and don't know how to proceed.

$\dfrac{27195^8-10887^8+10152^8}{26460}=\dfrac{3^5\cdot5^7\cdot7^{14}\cdot37^8-7013(2^6\cdot3^6\cdot47^2+3^2\cdot19^2\cdot191^2)(10152^4+10887^4)}{8}$

I'd like to ask, do you think this problem can be solved using only elementary methods?

Thanks in advance.

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