Wednesday
November 26, 2014

Homework Help: math

Posted by m on Thursday, July 23, 2009 at 7:18am.

can you answer this question in a different and more logical way than this method below:

we will examine the sum of cubes of two numbers, A aand B. Without losing generality, we will further assume that
A=2nX and
B=2n+kY
where
X is not divisible by 2
n is a positive integer and
k is a non-negative integer.

A3+B3
=(A+B)(A2-AB+B2)
=2n(X + 2kY) 22n(X2 - 2kXY + 22kY)
=23n(X + 2kY) (X - 2kXY + 22kY)
Thus A3+B3 has a factor 23n, but not 23n+1 since X is not divisible by 2.
Since 103n+1 requires a factor of 23n+1, we conclude that it is not possible that
103n+1=A3+B3


The question is prove that a number 10^(3n+1), where n is a positive integer, cannot be represented as the sum of two cubes of positive integers.


Thanks> please explain in full steps and in the most logical order

Thanks once again

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