This discussion board is a place to discuss our Daily Challenges and the math and science
related to those challenges. Explanations are more than just a solution — they should
explain the steps and thinking strategies that you used to obtain the solution. Comments
should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .

Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.

Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold

- bulleted - list

bulleted

list

1. numbered 2. list

numbered

list

Note: you must add a full line of space before and after lists for them to show up correctly

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 $171^{172}$.

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in`\(`

...`\)`

or`\[`

...`\]`

to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

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

Use binomial expansion:

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

Log in to reply

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

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

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

Log in to reply

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

Log in to reply

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

Log in to reply

also, it can solve by congruences

Log in to reply