Would much appreciate help to figure the remainder of 10^100mod 1001

[drama2]

Homework Statement

struggling to understand how to solve this question and would be extrememly grateful if someone who understood offered me a hand:

need to find the remainder of gogool divided by 1001. i.e 10^100mod1001. any assistance would be much welcomed as i have no hope figuring it out myself

Homework Equations

10^100mod1001

The Attempt at a Solution

 

[Wingeer]

What is the remainder of 10^3 when divided by 1001?

 

[drama2]

firstly thanks very much for responding. in regards to ur question the remainder would be a 1000. i assume you are implying that this is also the remainder for 10^100 mod 1001 though i can't figure out why that is the case and ift its not too much trouble would you please explain cos I am too stupid to be frank to see the connection :)

 

[SammyS]

firstly thanks very much for responding. in regards to ur question the remainder would be a 1000. i assume you are implying that this is also the remainder for 10^100 mod 1001 though i can't figure out why that is the case and ift its not too much trouble would you please explain cos I am too stupid to be frank to see the connection :)

I doubt that Wingeer is implying that. What he might be getting at is the following:

1000 ≡ -1 mod 1001
​

and

10¹⁰⁰ = 10·(10³)³³​

 

[drama2]

thanks for clarifyijng that SammyS. so to find the remainer from here using that result would the following be the correctway to do it.
10^3= 1000≡-1mod1001
therefore (10^3)^11=10^33=-1mod1001
(10^33)^3=10^99=1mod1001
therefore10^100=-10mod1001

from this would it be correct that the remainder is 10?


if anyone wants to answer this question for me i would sincerely appreciate it

 

[Wingeer]

It is correct. However I would have written it a bit different.
10^{1000}=10 \cdot (10^3)^{33} \equiv 10 \cdot (-1)^{33} = -10 (mod 1001)

 

[vela]

therefore10^100=-10mod1001

from this would it be correct that the remainder is 10?

No, that's not the correct remainder since 10≢-10 (mod 1001).

 

[SammyS]

Find a positive integer n, such that n < 1001 and n ≡ -10 mod 1001