The value \(b^n\) has both \(b\) and \(n\) as positive integers less than or equal to 15. What is the greatest number of positive factors \(b^n\) can have?
This is what I think gives you the greatest number of factors or divisors:
12^15 =2^30 x 3^15 =(30 + 1) x (15 + 1) =496 factors(divisors).
Hi guest answerer,
of the numbers from 1 to 15, 12 has the most factors so having b=12 sounds resonable.
n=15 also makes sense to me.
so maybe 12^15 is correct
BUT why do you say this has 496 factors?
Melody....I just read about the proof of this....roughly, it goes
Let n = p^a where p is a prime factor
Then...the divisors of n = p^a are
1, p, p^2, p^3, p^4.......p^a
So...the number of divisors = (a + 1)
p^a * q^b where p, q are prime factors
(a + 1) (b + 1) divisors