Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
Why is the last two digits of \(91^{71}\) the same as 91mod100? Similarly, why is it true that the last two digits of \(81^{81}\) the same as 81mod100, and that of \(71^{91}\) the same as 71mod100?
 9 months ago
 9 months ago
Why is the last two digits of \(91^{71}\) the same as 91mod100? Similarly, why is it true that the last two digits of \(81^{81}\) the same as 81mod100, and that of \(71^{91}\) the same as 71mod100?
 9 months ago
 9 months ago

This Question is Closed

RolyPolyBest ResponseYou've already chosen the best response.0
Maybe I should edit my question a bit. It should be "how do I know if the last two digits of \(91^{71}\) is the same as 91mod100? What if I need to find last two digits of\(91^{70}\)?
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
there the same thing, how much modular arithmetic do you know
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
I only know how to find x^y mod (n) using the trick that express y in binary number first, then take mod.
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
Why are you deleting your comment?
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
oh your refering to the binary exponentiation algorithm
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
This one: http://openstudy.com/updates/51165c4de4b09e16c5c86007
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$91^{71}\equiv 91 \text{ mod 100}$$ is what you are trying to show
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
Suppose the question is find the last two digits of \(91^{71}\). How would you start?
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
Do you know what the mod operator does?
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
if I were to write 91^71 in base 10
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
it would be $$91^{71}=a+b10+c10^2+d10^3...$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
now we can reduce that modulo 100, ok
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
Ehh!!! So, finding the last two digits of n is the same as finding n mod 100?
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
so $$91^{71}\text{ mod 100}=a+10b$$ Where a is are first digit and b are second digit
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
And finding the last n digits of m is the same as finding m mod (10^n)?
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
Which you can compute using your exponentiation by squaring algorithm or using a variety of other techniques
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
though for small numbers like yours, I wouldn't use that technique
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
What technique would you use?
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
Don't tell me calculator ~_~
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
well for $$91^{71}\text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
I would reduce the 91 modulo 100, so I get,
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$91^{71} \text{ mod 100} = (9)^{71} \text{ mod 100}= 9^{71} \text{ mod 100}$$
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
"reduce the 91 modulo 100"?
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
Yes, I think maybe you should read up a little on modular arithmetic,
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$91\equiv 9 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$91^{71}\equiv (9)^{71}\text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
so now we just have to compute 9 to the 71st power modulo 100
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
so its just computing $$9^{71}\text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
Now 9 is coprime to 100
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^{6}\equiv 1 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
raising both sides to the 12th power, gives
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^{72}\equiv 1 \text{ mod 100 }$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^{72}\equiv 1 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
but wqe want $$9^{71} \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
if we let $$9^{71}=x$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
we have $$9x\equiv 1 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
now all we have to do is solve for x modulo 100, and we have are solution, I know this looked like it took a while but its because I had to type over this chat, also there is many ways to solve somthing like this, if your just using a computer an algorithm like yours might be better.
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
But if your doing it by hand, its helpful to be familear with some of this stuff, because sometimes there are 'easyier' ways to do things.
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
I'll tag you if I have questions. I need to work it out myself to see if I really understand it. Sleep well :)
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
@Jack17 Problem problem problem "Now 9 is coprime to 100" "and ϕ(9)=6" " \(9^6≡1 mod 100\)" But isn't it \(a^{\phi(n)} ≡ 1 (\mod n)\) ? Why would we find ϕ(9)???
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
nice catch, this changes how you would proceed
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
Don't be so lazy... :(
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^{10}\equiv 1 \text{ mod 100 }$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
Which you can get by exponentiation by squaring
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
nvm im being stupid the original problem was $$9^{71} \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
this is the same as $$3^{142} \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
ahh maybe you should use your algorithm, i am to lazy to figure this out
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
142 is a LARGE number!!!
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
then use your technique
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
use the fact $$9^{10}\equiv 1 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
so $$9^{70}\equiv 1 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^{71}\equiv 9\text{ mod 100}$$
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
9^(40) ≡1 mod 100 since phi(100) is 40 o_o
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^{71}\equiv 9 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^{71}\equiv 91 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$91^{71}\equiv 91 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
the first two digits of $$91^{71}$$ are 1 and 9
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
9^(10) ≡1 mod 100 < How do you work this out?
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^2\equiv 19 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^4\equiv 19^2 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^4\equiv 39 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^8 \equiv 39^2 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^8\equiv 21 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^{10}\equiv 21*9^2 \text{ mod 100}$$
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
$$9^{10}\equiv 1 \text{ mod 100}$$
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
Got it! Have never thought that it works in this way... Another problem is that how you can draw the conclusion "so \( 9^{70}≡1 \mod 100\)" from \(9^{10}≡1 \mod 100\) quickly?
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
I raised both sides to the 7th power
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
Ok, I didn't notice that, sorry!!
 9 months ago

Jack17Best ResponseYou've already chosen the best response.1
If $$a\equiv b \text{ mod m}$$ and $$c\equiv d \text{ mod m}$$ then $$ac\equiv bd \text{ mod m}$$
 9 months ago

RolyPolyBest ResponseYou've already chosen the best response.0
Thanks!!! It looks pretty good now :) Btw, if you want to type LaTeX in the same line as the words, use \( instead of \[ . Thanks again for your help :)
 9 months ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.