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Mkay, so factor it...
(2⁴)(3)(5²)
This means that it has ((4+1)(1+1)(2+1)) factors, or 30 factors. So what now?
Well, here's a little trick. To find the number of odd factors, simply divide the number of total factors by the exponent of the 2's PLUS one.
30/(4+1) = 6
So 6 odd factors, 24 even factors.
As to why this works, think about it. All you are doing is taking the largest possible odd factor of 1200 and finding the number of factors in that.
1200 = (2⁴)(75)
And basically, you're just finding the number of factors in 75. Anything that includes a 2 in there would make it even, so it's irrelevant.
(2⁴)(3)(5²)
This means that it has ((4+1)(1+1)(2+1)) factors, or 30 factors. So what now?
Well, here's a little trick. To find the number of odd factors, simply divide the number of total factors by the exponent of the 2's PLUS one.
30/(4+1) = 6
So 6 odd factors, 24 even factors.
As to why this works, think about it. All you are doing is taking the largest possible odd factor of 1200 and finding the number of factors in that.
1200 = (2⁴)(75)
And basically, you're just finding the number of factors in 75. Anything that includes a 2 in there would make it even, so it's irrelevant.
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A factor is even if the power of 2 is non-zero. An even factor must take the form:
2^a * 3^b * 5^c, with 1 <= a <= 4, 0 <= b <= 1, and 0 <= c <= 2.
There are 4 possible values of a, 2 possible values of b, and 3 possible values of c. Thus, there are 4*2*3 = 24 even factors.
You know that there are 30 factors. Since there are 24 even factors, the rest of them are odd. Thus, there are 30 - 24 = 6 odd factors.
I hope this helps!
2^a * 3^b * 5^c, with 1 <= a <= 4, 0 <= b <= 1, and 0 <= c <= 2.
There are 4 possible values of a, 2 possible values of b, and 3 possible values of c. Thus, there are 4*2*3 = 24 even factors.
You know that there are 30 factors. Since there are 24 even factors, the rest of them are odd. Thus, there are 30 - 24 = 6 odd factors.
I hope this helps!