ABSTRACT ALGEBRA problem Need help fast Find the order of 8

ABSTRACT ALGEBRA problem. Need help fast!

Find the order of (8, 10) in Z_12 times Z_18.

Solution:

Order of a product of disjoint cycles is least common multiple of the orders of the cycles

Order of 8 in Z12 = 12/gcd(8, 12) = 12/4 = 3.

Order of 10 in Z18 = 18/gcd(10, 18) = 18/2 = 9.

So the order of (8, 10)

in Z12 × Z18

is lcm(3, 9) = 9.

Hence over.