Prove Divisibility:

Question

Prove that

 

(a + b + c)³³³ - a³³³ - b³³³ - c³³³ 

 

is divisible by

 

(a + b + c)³ - a³ - b³ - c³

 

How can I start this? 

 

Thanks.

Answer 1

We could divide the first polynomial by the second polynomial using long division.  If the remainder is zero, then it is divisible. 

 

Another way to look at it is that both polynomials have the same variables.  They both have the sum of a, b, and c term.  They both also have  -aⁿ - bⁿ - cⁿ  term, where n=3 and 333.

 

The numbers 3 and 333 have a common factor of 3, and 333 is divisible by 3. 

 

333 ÷ 3 = 111

 

 

Therefore, if we were to divide, which will be too difficult to do and messy, we know the remainder will be zero.