Question

Determine n between 0 and 8 such that 1111^2222 = n mod 20

Answer 1

take the help of log

Answer 2

i would start by reducing 1111 to 11

Answer 3

help of log? I tried to log both sides and could do that with the left side but ran into problems with the right.

Answer 4

since the gcd of 1111 and 20 is 1, they are relatviely prime; so we can determine the value of phi(20) to reduce the exponent with

Answer 5

since \[a^{\phi(n)}\cong1(mod~n)\]when (a,n)=1

Answer 6

phi(20) counts all the residue classes that are relatively prime to 20, mod20 ...

phi(20) = 8

8 divides 2222 how many times? with what left over?

Answer 7

277 times with 0.75 left over. So multiply 0.75 times 9?

Answer 8

you keep confusing remainder with decimal

Answer 9

2222 mod 8 = 6 since

2222 = 8(277) + 6

Answer 10

Oh I see...by doing long division?

Answer 11

yes, that is one way to accomplish it

Answer 12

Okay, I understand that. Thank you for clarifying!

Answer 13

the other way is to divide 2222 by 8 and ignore the decimal; then you know that 8(277) is close to 2222; and see that you have to add 6 to get all the way there

Answer 14

so, \[1111^{2222}=n~(20)\]
\[1111^{\phi(20)(277)+6}=n~(20)\]
\[1111^{\phi(20)(277)}*1111^6=n~(20)\]
\[1*1111^6=n~(20)\]
\[1111^6=n~(20)\]

that at least makes it less daunting

Answer 15

so, as satellite suggested, reduce 1111; what is 1111 mod 20?

Answer 16

It would be 55 R 11 Right?

Answer 17

looks good, and its the R that we want: so 1111 mod 20 = 11

notice that we have 1111^6

1111.1111.1111.1111.1111.1111 = n (20)

we can replace all the 1111 with their 11 equals

11.11.11.11.11.11 = n (20)

11^6 = n (20) getting a little easier to read yet?

Answer 18

Much! So basically now we can solve the problem in the regular way since it is in small enough of a form?

Answer 19

yes, now it reduced to a workable form

Answer 20

Would the answer be 88578 R1. So 1?

Answer 21

id have to work it out to be sure :)

Answer 22

(11.11).(11.11).(11.11) = n (20)

121.121.121 = n (20)

121 = 20(6) + 1; R 1

1.1.1 = n (20)

1 = n (20) i believe it is, but id better chk the wolf to be certain

Answer 23

Okay that sounds good! You are a huge help! This problem was rather daunting to me :) You put it in more understandable terms for sure!

Answer 24

http://www.wolframalpha.com/input/?i=1111%5E%282222%29%3Dn%28mod20%29

we either did something spectacularly right, or just got lucky ;) n=1

Answer 25

Wow! Well either way, I'm perfectly content :) Thanks again for your help! It was a real blessing!

Answer 26

youre welcome, i took number theory this term and its almost over; so that was fun ;)

good luck

Answer 27

You too!! :)