Please help me in this problem.

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

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Comments

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

Use binomial expansion:

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

gopinath no - 7 years, 8 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 $171^{172}$ .

Michael Tong - 7 years, 8 months ago

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

Sayan Chaudhuri - 7 years, 5 months ago

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

Joefer Guillermo - 7 years, 8 months ago

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

Sayan Chaudhuri - 7 years, 8 months ago

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

Fahad Shihab - 7 years, 8 months ago

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

Silas Hundt Staff - 7 years, 8 months ago

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

gopinath no - 7 years, 8 months ago

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

Utsav Singhal - 7 years, 8 months ago

also, it can solve by congruences

jinay patel - 7 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$