How many factors of 2^3 * 3^4 * 5^5 are even numbers?

A. 20

B. 30

C. 90

D. 100

E. 120

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