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

Note by Joefer Guillermo
6 years, 10 months ago

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

Use binomial expansion:

\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}

- 6 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 $171^{172}$.

- 6 years, 10 months ago

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

- 6 years, 7 months ago

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

- 6 years, 10 months ago

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

- 6 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!

- 6 years, 10 months ago

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

Staff - 6 years, 10 months ago

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

- 6 years, 10 months ago

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

- 6 years, 10 months ago

also, it can solve by congruences

- 6 years, 10 months ago