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

@Raja Metronetizen
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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
·
3 years, 10 months ago

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also, it can solve by congruences
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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

## Comments

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

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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– Michael Tong · 3 years, 10 months ago

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 \(171^{172}\).Log in to reply

– Raja Metronetizen · 3 years, 8 months ago

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

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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– Gopinath No · 3 years, 10 months ago

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

– Silas Hundt Staff · 3 years, 10 months ago

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

– Raja Metronetizen · 3 years, 10 months ago

This problem is also for my computer..Don't know why?Log in to reply

– Fahad Shihab · 3 years, 10 months ago

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

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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