1. ## Length of 3

For any positive integer n, n > 1, the "length" of n is the number of positive primes (not necessarily distinct) whose product is n. For example, the length of 50 is 3 since 50 = (2)(5)(5). Given this information, which of the following integers has a length of 3?

A. 3
B. 15
C. 60
D. 64
E. 105

I said 15  2  2 = 30  2 = 60.

2. ## Re: Length of 3 Originally Posted by harpazo For any positive integer n, n > 1, the "length" of n is the number of positive primes (not necessarily distinct) whose product is n. For example, the length of 50 is 3 since 50 = (2)(5)(5). Given this information, which of the following integers has a length of 3?

A. 3
B. 15
C. 60
D. 64
E. 105

I said 15  2  2 = 30  2 = 60.
60 doesn't work because it factors as 2^2 x 3 x 5. You need them to be prime factors. 15 is not prime.

Now 105 = 5^3 = 5 x 5 x 5 so this would have a length of 3.

-Dan

3. ## Re: Length of 3 Originally Posted by harpazo For any positive integer n, n > 1, the "length" of n is the number of positive primes (not necessarily distinct) whose product is n. For example, the length of 50 is 3 since 50 = (2)(5)(5). Given this information, which of the following integers has a length of 3?

A. 3
B. 15
C. 60
D. 64
E. 105

I said 15 • 2 • 2 = 30 • 2 = 60.
Look at this tool. Using that tool we see that the length of 168 is five (3+1+1=5)
You can simple 168 to any other number, hit enter then proceed. Explore that site for other goodies.

4. ## Re: Length of 3 Originally Posted by topsquark 60 doesn't work because it factors as 2^2 x 3 x 5. You need them to be prime factors. 15 is not prime.

Now 105 = 5^3 = 5 x 5 x 5 so this would have a length of 3.

-Dan
Isn't 5^3 = 125?

5. ## Re: Length of 3 Originally Posted by Plato Look at this tool. Using that tool we see that the length of 168 is five (3+1+1=5)
You can simple 168 to any other number, hit enter then proceed. Explore that site for other goodies.
I use Wolfram occasionally.

6. ## Re: Length of 3 Originally Posted by harpazo I use Wolfram occasionally.
If I were you, given your poor foundation in basic mathematics, I would use it all the time.

7. ## Re: Length of 3 Originally Posted by Plato If I were you, given your poor foundation in basic mathematics, I would use it all the time.
I'll try using Wolfram as much as possible. I am not in a math class. No need to stress it out. I find math interestingly fun to play with.

8. ## Re: Length of 3 Originally Posted by topsquark 60 doesn't work because it factors as 2^2 x 3 x 5. You need them to be prime factors. 15 is not prime.

Now 105 = 5^3 = 5 x 5 x 5 so this would have a length of 3.

-Dan
For example, does 8 have a length of 3 considering that 2^3 = 2 • 2 • 2?