How many factors of 2^3 * 3^4 * 5^5 are even numbers?
A. 20
B. 30
C. 90
D. 100
E. 120
Solution:
The easier question to answer is "how many factors of 2^3 * 3^4 * 5^5 are NOT even numbers."
Those factors will only have prime factors of 3 or 5. To construct a factor that is odd, there are 5 possible powers of 3 (0, 1, 2, 3, 4) and 6 possible powers of 5 to choose from (0, 1, 2, 3, 4, 5). Therefore there are 5 * 6 = 30 factors that are not even.
With the same theory, there are 4 * 5 * 6 = 120 factors in total. Subtracting the 30 odd factors, there are 90 even.
Ans: C
A. 20
B. 30
C. 90
D. 100
E. 120
Solution:
The easier question to answer is "how many factors of 2^3 * 3^4 * 5^5 are NOT even numbers."
Those factors will only have prime factors of 3 or 5. To construct a factor that is odd, there are 5 possible powers of 3 (0, 1, 2, 3, 4) and 6 possible powers of 5 to choose from (0, 1, 2, 3, 4, 5). Therefore there are 5 * 6 = 30 factors that are not even.
With the same theory, there are 4 * 5 * 6 = 120 factors in total. Subtracting the 30 odd factors, there are 90 even.
Ans: C
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