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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} · 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 $$171^{172}$$. · 3 years, 10 months ago

thanks ...I have now learnt how to solve these kinds of problems....thanks again... · 3 years, 8 months ago

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

Even I can't see it (in chrome)! Preview was fine, and firefox displays it. · 3 years, 10 months ago

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

This problem is also for my computer..Don't know why? · 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! · 3 years, 10 months ago

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