I have a solution and I compared it to the solution we were given. I have bolded where there is a difference. I don't know why this is incorrect:
Count the elements of order $10$ in $D_{10} \times \mathbb{Z}_{450}$
My solution: The order of an element in a direct product of groups is the lcm of each component. To obtain order $4$ we can have
- First component $1,2,5$ or $10$, second Component $10$ - this gives $4*(1+11+4+4) = 80$ elements
- Second component $1,2,5$ or $\mathbf{10}$, second Component $10$ - this gives $4*(1+1+4+4) = 40$ elements
- First Component $5$, first component $2$ - this gives $4*1=4$ elements
- First Component $2$, first component $5$ - this gives $11*4=44$ elements
Thus 168 elements of order 10.
Why aren't the order $10$ elements included in the second bullet point?
The second bullet should not count the elements where the first and second component both have order 10, since you already included them in the first bullet.