Please help me in this problem.

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

Note by Joefer Guillermo
6 years ago

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2 votes

  Easy Math Editor

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171172=(1+170)172171^{172} = (1+170)^{172}

Use binomial expansion:

1711721+172170+(1722)1702(mod1000)1+240+400(mod1000)641(mod1000) \begin{aligned} 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{aligned}  

gopinath no - 6 years 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(170)^3) so they have no bearing on the last three digits of 171172171^{172}.

Michael Tong - 6 years ago

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

Sayan Chaudhuri - 5 years, 10 months ago

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

Joefer Guillermo - 6 years ago

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

Sayan Chaudhuri - 6 years 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 - 6 years 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 - 6 years ago

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

gopinath no - 6 years ago

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

Utsav Singhal - 6 years ago

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also, it can solve by congruences

jinay patel - 6 years ago

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