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# The value has both and as positive integers less than or equal to 15. What is the greatest number of positive factors &nbs

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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?

Oct 14, 2018
edited by Guest  Oct 14, 2018

#1
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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).

Oct 14, 2018
edited by Guest  Oct 14, 2018
#2
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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
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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
#4
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ok thanks Chris :)

Melody  Oct 15, 2018