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Please help me in this problem.

What is the last 3 digits of 171^{172}?

Note by Joefer Guillermo
4 years, 1 month ago

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2 votes

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\(171^{172} = (1+170)^{172}\)

Use binomial expansion:

\( \begin{align} 171^{172} & \equiv 1+172\cdot 170 +\binom{172}{2} 170^2 \pmod {1000}\\ & \equiv 1+240+400\pmod {1000}\\ & \equiv 641\pmod {1000} \end{align}  \)

Gopinath No - 4 years, 1 month ago

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In case people are confused, the rest of the terms in the expansion are multiples of 1000 (since they contain a factor of \((170)^3\)) so they have no bearing on the last three digits of \(171^{172}\).

Michael Tong - 4 years, 1 month ago

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thanks ...I have now learnt how to solve these kinds of problems....thanks again...

Raja Metronetizen - 3 years, 11 months ago

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why can't I see the solution Gopinath commented? Is there a problem with my computer?

Joefer Guillermo - 4 years, 1 month ago

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Even I can't see it (in chrome)! Preview was fine, and firefox displays it.

Gopinath No - 4 years, 1 month ago

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This is a bug in our math rendering system. We are looking into it now—thanks for mentioning it!

Silas Hundt Staff - 4 years, 1 month ago

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This problem is also for my computer..Don't know why?

Raja Metronetizen - 4 years, 1 month ago

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the computer will 'choke' because of the calculation. We need an advanced algorithm and wait for minutes till the computer calculates it!

Fahad Shihab - 4 years, 1 month ago

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also, it can solve by congruences

Jinay Patel - 4 years, 1 month ago

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yes,by using binomial expansion we get ans...641

Utsav Singhal - 4 years, 1 month ago

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