Order of a as an element of group G

[kiddopop]

For element a in the group G, given below, find the order of a as an element of G. (If possible, answer without listing all elements of the cyclic subgroup generated by x, but explain how you know your answer is correct.)
a=(2,5) in Zsub 6 x sub 15.



I know how to find an order of a with just a set of integers, but when it's Zsub 6 x sub 15, I just have no idea.

 

[daon]

The order of an element is the number of times you perform the operation on that element to get to the identity in the group. In this case it is under componentwise addition and your identity is (0 mod 6,0 mod 15).

Hint:
6/2 = 3
15/5 = 3

Try this also for (4,5) -- it will be the same order as (2,5).. why? Then the result for (5,2) is quite different, surprised? Try to come up with a formula.