Hello. I'm trying to understand the explanation for this problem here.
Problem 110 in the OG 12 states:
If p is the product of the integers from 1 to 30 inclusive, what is the greatest integer k for which 3^k is a factor of p?
Answer:
3
6 = 2 x 3
9 = 3 x 3
12 = 2 x 2 x 3
15 = 3 x 5
18 = 2 x 3 x 3
21 = 3 x 7
24 = 2 x 2 x 2 x 3
27 = 3 x 3 x 3
30 = 2 x 3 x 5
Therefore the answer is 3^14 because there are 14 factors of 3 above. What principle states that you can multiply 3 to the power of the number of factors? Is there any other way to solve this problem that's easier? Any assistance is greatly appreciated.
Problem 110 in the OG 12 states:
If p is the product of the integers from 1 to 30 inclusive, what is the greatest integer k for which 3^k is a factor of p?
Answer:
3
6 = 2 x 3
9 = 3 x 3
12 = 2 x 2 x 3
15 = 3 x 5
18 = 2 x 3 x 3
21 = 3 x 7
24 = 2 x 2 x 2 x 3
27 = 3 x 3 x 3
30 = 2 x 3 x 5
Therefore the answer is 3^14 because there are 14 factors of 3 above. What principle states that you can multiply 3 to the power of the number of factors? Is there any other way to solve this problem that's easier? Any assistance is greatly appreciated.
