Analysis:
To find the number of elements of order 15 in the additive group Z50*Z60, we need to understand the structure of the group and the concept of order.

Structure of the Group:
The additive group Z50*Z60 consists of pairs of integers (a, b), where a is an element of Z50 and b is an element of Z60. The group operation is defined as (a, b) + (c, d) = (a + c, b + d) (mod 50, 60).

Order of an Element:
The order of an element in a group is the smallest positive integer n such that the element raised to the power of n gives the identity element of the group. In this case, the identity element is (0, 0).

Solution:
To find the elements of order 15, we need to consider the pairs (a, b) such that (a, b)^15 = (0, 0).

Step 1: Finding Elements of Order 15 in Z50:
For an element (a, b) to have order 15, we need (a, b)^15 = (0, 0) (mod 50, 60).

Since (a, b) is in Z50*Z60, (0, b) + (a, 0) = (0, 0) (mod 50, 60).

This implies a = 0 (mod 50) and b = 0 (mod 60).

So, the possible values of a are multiples of 50 in the range [0, 49] and the possible values of b are multiples of 60 in the range [0, 59].

Step 2: Counting the Elements:
The number of elements of order 15 in Z50*Z60 is the number of possible pairs (a, b) satisfying the conditions from Step 1.

The number of possible values of a is 50 (multiples of 50 in the range [0, 49]).

The number of possible values of b is 60 (multiples of 60 in the range [0, 59]).

Therefore, the total number of elements of order 15 in Z50*Z60 is 50 * 60 = 3000.

Conclusion:
The number of elements of order 15 in the additive group Z50*Z60 is 3000.