
Glossary: Prime Pages: Top 5000: 
In a group (a special set with an operation on it like addition or multiplication), elements have orders. Usually, on these pages, the group is the set of nonzero remainders modulo a prime and the order of a modulo p then is the least positive integer n such that a^{n} 1 (mod p).
For example, let us use a=3 and p=7. Look at the powers of 3 modulo 7: 3^{1} 3, 3^{2} 2, 3^{3} 6, 3^{4} 4, 3^{5} 5, 3^{6} 1The order of 3 modulo 7 is 6. The order of 2 modulo 7 is 3. The order of 6 modulo 7 is 2. When working modulo a prime, the set of nonzero remainders form a multiplicative group. This is not true modulo a composite. Fermat's Theorem tells us the order of a nonzero element modulo a prime divides the prime minus one. Euler's theorem gives us a similar result for composites.
Chris K. Caldwell © 19992017 (all rights reserved)
