Question

What is \(11^{644}(mod~~645)\)
Please, help

Answer 1

Yes, I am here. :)

Answer 2

645 = 3*5*43

Answer 3

Notice below :

\(11^{644} \equiv (-1)^{644}\equiv 1 \pmod{3}\)

\(11^{644} \equiv (1)^{644}\equiv 1 \pmod{5}\)

\(11^{644} \equiv (11)^{42*15+14}\equiv 11^{14}\equiv 1 \pmod{43}\)

Answer 4

that means
\(3 \mid (11^{644}-1)\)
\(5 \mid (11^{644}-1)\)
\(43 \mid (11^{644}-1)\)

since \(3, 5, 43\) are primes, it follows
\(3*5*43 \mid (11^{644}-1)\)
which is same as saying \[11^{644}\equiv 1\pmod{3*5*43}\]

Answer 5

Thank you so much. My question is how we know 11^(14) =1 mod 43 without calculator?

Answer 6

that is a good question...
before that, did you get why \(11^{42*15} \equiv 1\pmod{43}\) ?

Answer 7

Yes

Answer 8

may i know why you think that reduces to 1 ?

Answer 9

because 43 is a prime, and (43,11) =1 hence \(11^{42}\equiv 1(mod 43)\) and that leads to \(11^ {42*15}\equiv 1(mod 43)\)

Answer 10

Fermat told me that :)

Answer 11

you shouldn't talk to the dead

Answer 12

He "told" me through the book. :)

Answer 13

lol nice, do you feel \(11^{14}\) is hard to reduce ?

Answer 14

Yes.

Answer 15

how ?

Answer 16

show me how it is hard

Answer 17

because, i have figured out the answer in 2-3 steps working it in my head...

Answer 18

11^14 = 11*11*11*........(14 times) . I don't know how to make it easier.

Answer 19

Alright, here it is :

\[11^{14} \equiv 121^7 \equiv (-8)^7\equiv - 2^{3*7}\equiv -128^3 \equiv -(-1)^3 \equiv 1\pmod{43}\]

Answer 20

oh, it is much easier. Thank you so much. I got it.