Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.
Thanks for anything you can add to the picture.
this seems like a follow up to this thread. you should have asked this question there
powers of 2 would yield the smallest number. since powers of 1 would only yield 1 and thus would not reach 4096, and powers of 3 (or more, or a product with numbers greater than 2) would be greater than corresponding powers of 2 and so would not be minimum
Hello, reagan3nc!
There is a theorem that might help explain it.Why is 4096 the smallest number with 13 factors?
What does 2^n have to do with it?
I understand in order to get an odd number of factors you have to use a perfect square.
Given: . . (prime factorization)
. . the number of factors of is: .
. . . . Add one to each exponent and multiply.
Example: .
. . . factors.
We want a number so that: .
Hence, has a prime factorization with an exponent of 12: .
. . And the smallest occurs when
Got it?