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What is the largest positive integer n for which n^3+100 is divisible by n+1?

 Jun 21, 2021

Best Answer 

 #1
avatar+179 
+2

Because we can factorize n^3+1, we simply write n^3+100 as n^3+1+99.  

 

Thus, because n+1 divides n^3+100, aka n^3+1+99, n+1 divides (n+1)(n^2-n+1)+99.  

 

Knowing that (n+1) will always divide (n+1)(n^2-n+1), we simply need n+1 to divide 99.  

 

The largest divisor of 99 is 99, meaning that n+1 will equal 99 to maximize n.  With this, we have n=98.  

 

 

 

The answer is 98.

 Jun 21, 2021
 #1
avatar+179 
+2
Best Answer

Because we can factorize n^3+1, we simply write n^3+100 as n^3+1+99.  

 

Thus, because n+1 divides n^3+100, aka n^3+1+99, n+1 divides (n+1)(n^2-n+1)+99.  

 

Knowing that (n+1) will always divide (n+1)(n^2-n+1), we simply need n+1 to divide 99.  

 

The largest divisor of 99 is 99, meaning that n+1 will equal 99 to maximize n.  With this, we have n=98.  

 

 

 

The answer is 98.

EnchantedLava68 Jun 21, 2021
 #2
avatar+120930 
0

Very nice,  EL   !!!!!

 

 

 

cool cool cool

CPhill  Jun 21, 2021

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