Show that 5448 has order 5^4 in Z/11251Z. How do I show this quickly, I know that all elements must have an order that divides order(Z/11251) according to lagrange's theorem but how do i pinpoint that it is 5^4 and not 5^3 or 5^2?
Asked
Active
Viewed 58 times
1 Answers
0
If you calculate that $$ 5448^{5^4} \equiv 1 \pmod{11251} \quad\text{and}\quad 5448^{5^3} \not\equiv 1 \pmod{11251}, $$ then that is enough to prove that the order of $5448$ modulo $11251$ equals $5^4$. Do you see why?
Those calculations can be done quickly with fast modular exponentiation - hopefully that's something you've been shown.
Greg Martin
- 92,241
-
I was doing this but 5448^(5^4) is not that easy to calculate? – ugradmath Dec 19 '14 at 09:29
-
Not as an integer, no, but modulo 11251, the calculation is indeed easy. http://en.wikipedia.org/wiki/Modular_exponentiation – Greg Martin Dec 19 '14 at 19:06