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In terms of N, a natural number, how many more positive integral factors does \(24^n\) than \(16^n \)?

 

What I did:

 

24^n is now 3^n * 2^3n

 

16^n is now 2^4n

 

We can find the number of factors in a number by prime factoring it, then taking each prime factor's exponent and adding 1 to it, then mulitply it with the others.

 

So 24^n has 3n^2+4n+1 factors

 

And 16^n has 4n+1 factors

 

This means that there are 3n^2 more factors.

 

Am i Right?

 May 31, 2019
 #1
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YES! Very good. You can always confirm it by using actual powers such as:16^5  vs  24^5. And the difference is:3 x 5^2.

 May 31, 2019
 #2
avatar+2856 
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Thank you!

 May 31, 2019
 #3
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Very nice explantation!

 May 31, 2019

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