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

Note by Joefer Guillermo
6 years ago

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.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $\boxed{123}$

Sort by:

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

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 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 ago

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

- 5 years, 10 months ago

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

- 6 years ago

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

- 6 years 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 ago

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

Staff - 6 years ago

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

- 6 years ago

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

- 6 years ago

also, it can solve by congruences

- 6 years ago