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

Note by Joefer Guillermo 4 years, 1 month ago

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

Use binomial expansion:

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

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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 \((170)^3\)) so they have no bearing on the last three digits of \(171^{172}\).

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

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

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

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

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

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

also, it can solve by congruences

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

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## Comments

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

Use binomial expansion:

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

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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 \((170)^3\)) so they have no bearing on the last three digits of \(171^{172}\).

Log in to reply

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?

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

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

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