I Don't Get It!

I came across a question which uses Fermat's little theorem:

The Question:
Find a positive integer n such that $7n^{25}$ - 10 is divisible by 83.

The Solution given in the book:
Since 7 x 37 = 259 = 10 mod 83
We have to find a value of n such that $7n^{25}$ = 7 x 37 mod 83
This is equivalent to $n^{25}$ = 37 = $2^{20}$ mod 83
By Fermat's Theorem,
$2^{82k}$ = 1 mod 83 for all k.
So we need to choose n such that $n^{25}$ = $2^{82k+20}$ mod 83
This will be satisfied if k=15
Therefore $n^{25}$ = $2^{1250}$ mod 83
And so n = $2^{50}$
This gives one value.

My problem is that I don't understand the fourth line

That is how 37 = $2^{20}$ mod 83?

So can someone please explain this?

#NumberTheory

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:

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
paragraph 1 paragraph 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"

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}`	$\boxed{123}$

Comments

Sort by:

Top Newest

$2^8 \equiv 256 \equiv 249 + 7 \equiv 7$ (mod $83$ )
$\Rightarrow 2^{16} \equiv 7^2$ (mod $83$ )
$\Rightarrow 2^{20} \equiv 49 \cdot 16 \equiv 784 \equiv 37$ (mod $83$ )

Ameya Daigavane - 4 years, 4 months ago

Wow !! I didn't think of that. Thanks a LOT Sir !!!

abc xyz - 4 years, 4 months ago

Note that 2 is a primitive root modulo 83, so that even if the remainder weren't 37, we could find another exponent (but finding the exact exponent is difficult, in general.) instead of 20 here.

Ameya Daigavane - 4 years, 4 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}`	$\boxed{123}$