Posted by John on Sunday, March 10, 2013 at 4:46pm.
Find the smallest positive integer N such that 13^N \equiv 1 \pmod{2013}.

Algebra  Count Iblis, Sunday, March 10, 2013 at 7:20pm
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
Answer This Question
Related Questions
 Algebra  what is the smallest positive integer N such that 13^N \equiv 1 \pmod{...
 math  Let S={1,2,3,4,…,2013} and let n be the smallest positive integer such ...
 Algebra  Find the smallest positive integer N \neq 23 such that the fraction \...
 Algebra  Find the smallest positive integer N \neq 23 such that the fraction \...
 arithmetic  Find the smallest positive integer P such that the cube root of 400...
 math  there are three consecutive positive integers such that the sum of the ...
 Math  Let a be an integer, then there are integers X, Y such that aX+(a+1)Y=1. ...
 algebra  find three consecutive odd integers such that the sum of the middle ...
 Math  Find the smallest positive integer d such that d=105m+216n, where m & n ...
 Math  Find the smallest positive integer n such that the equation 455x+1547y=50...
More Related Questions