Good stuff, Mike.
Obviously it's critical to not solve this one by determining all 30 factors and then reducing that set to the odd values. (Exercise: if you don't know how to quickly determine that there are 30 factors of 1200, do some drills!)
So, as you say, the key to these is deriving a complete set of prime factors.
Here's my approach:
Get the prime factors of 1200:
1200 = 12*100 = 2*2*3*2*5*2*5 = 2^4 * 3^1 * 5^2
Now, utilize the multiplication rules:
odd*odd = odd
even*odd = even
So, the number of odd factors can be determined by calculating the number of factors when x=0 for 2^x. Why? Because then we are left with only odd*odd.
In this case that means we want the number of factors of 2^0 * 3^1 * 5^2, which is (0+1)(1+1)(2+1) = 1*2*3 = 6.
Answer A
Also, consider this approach:
Given a prime factorization of 2^x * 3^y * 5^z, then the number of odd factors will be:
(total number of factors)/(x+1)
Why is it so? (It's just another format of the first approach)
Try it in this problem:
(4+1)(1+1)(2+1)/(4+1) = 6
Answer A
