Question:
Let a be an element of a group G and O(a) = 35, then O(a^15) is ____________

Answer:
To find the order of the element a^15 in the group G, we need to understand the concept of the order of an element in a group.

Order of an Element:
The order of an element a in a group G, denoted as O(a), is the smallest positive integer n such that a^n = e, where e is the identity element of the group. In other words, the order of an element is the smallest power of the element that gives the identity element.

Key Points:
- The order of an element is always a positive integer.
- The order of the identity element is always 1.
- The order of an element a is equal to the order of its cyclic subgroup generated by a.

Given Information:
- O(a) = 35

Using the Key Points:
- Since the order of an element is always a positive integer, O(a) = 35 implies that a^35 = e.
- The order of a^15 is the smallest positive integer n such that (a^15)^n = e.

Calculating the Order of a^15:
- Let's assume the order of a^15 is m.
- (a^15)^m = e
- a^(15m) = e
- Since the order of a is 35, we know that a^35 = e.
- Therefore, (a^15)^m = a^(35k), where k is an integer.
- Simplifying, we get a^(15m) = a^(35k)
- Since a^(15m) = a^(35k), we can conclude that 15m ≡ 35k (mod 35)
- Dividing both sides by 5, we get 3m ≡ 7k (mod 7)
- The smallest positive integer solution for this congruence equation is m = 7.

Conclusion:
Therefore, the order of a^15 in the group G is 7.