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?

Guest Oct 14, 2018

edited by
Guest
Oct 14, 2018

#1**+1 **

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).

Guest Oct 14, 2018

edited by
Guest
Oct 14, 2018

#2**+1 **

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
Oct 15, 2018

#3**+2 **

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)

So

p^a * q^b where p, q are prime factors

has

(a + 1) (b + 1) divisors

CPhill
Oct 15, 2018