I was doing a question yesterday on very elementary number theory, and I came across this pattern.
$2^1=2 \pmod {13}$
$2^2=4 \pmod {13}$
$2^3=8 \pmod {13}$
$2^4=3 \pmod {13}$
$2^5=6 \pmod {13}$
$2^6=12 \pmod {13}$
$2^7=11 \pmod {13}$
$2^8=9 \pmod {13}$
$2^9=5 \pmod {13}$
$2^{10}=10 \pmod {13}$
$2^{11}=7 \pmod {13}$
$2^{12}=1 \pmod {13}$
$2^{13}=2 \pmod {13}$
I noticed the remainder always is unique for every block multiples of powers 13. And it repeats!.
Also I tried this for base 3 and base 4 base 5. All showed similar patterns.
Why?