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Please help me in this problem.

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

Note by Joefer Guillermo
3 years, 10 months ago

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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 · 3 years, 10 months ago

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@Gopinath No 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 · 3 years, 10 months ago

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@Gopinath No thanks ...I have now learnt how to solve these kinds of problems....thanks again... Raja Metronetizen · 3 years, 8 months ago

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why can't I see the solution Gopinath commented? Is there a problem with my computer? Joefer Guillermo · 3 years, 10 months ago

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@Joefer Guillermo Even I can't see it (in chrome)! Preview was fine, and firefox displays it. Gopinath No · 3 years, 10 months ago

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@Joefer Guillermo This is a bug in our math rendering system. We are looking into it now—thanks for mentioning it! Silas Hundt Staff · 3 years, 10 months ago

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@Joefer Guillermo This problem is also for my computer..Don't know why? Raja Metronetizen · 3 years, 10 months ago

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@Raja Metronetizen the computer will 'choke' because of the calculation. We need an advanced algorithm and wait for minutes till the computer calculates it! Fahad Shihab · 3 years, 10 months ago

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also, it can solve by congruences Jinay Patel · 3 years, 10 months ago

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yes,by using binomial expansion we get ans...641 Utsav Singhal · 3 years, 10 months ago

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