Algebra
posted by John .
Find the smallest positive integer N such that 13^N \equiv 1 \pmod{2013}.

2013 = 3*11*61
phi(2013) = 2*10*60 = 2^4*3*5^2
13^[phi(2013)/2] = 1
13^[phi(2013)/3] = 562
13^[phi(2013)/5] = 1
This means that
13^[phi(2013)/10] = 1
Dividing the exponent by 5 gives:
13^[phi(2013)/50] = 1831
So, we only need to try dividing the exponent of phi(2013)/10 by factors of 2 to see if we still get 1:
13^[phi(2013)/20] = 1
13^[phi(2013)/40] = 1
13^[phi(2013)/80] = 1
So, the order of 13 is phi(2013)/40 = 30