Maths
posted by Matt .
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.
thanx

We will examine the sum of cubes of two numbers, A and B. Without losing generality, we will further assume that
A=2^{n}X and
B=2^{n+k}Y
where
X is not divisible by 2
n is a positive integer and
k is a nonnegative integer.
A^{3}+B^{3}
=(A+B)(A^{2}AB+B^{2})
=2^{n}(X + 2^{k}Y) 2^{2n}(X^{2}  2^{k}XY + 2^{2k}Y²)
=2^{3n}(X + 2^{k}Y) (X²  2^{k}XY + 2^{2k}Y²)
Thus A^{3}+B^{3} has a factor 2^{3n}, but not 2^{3n+1} since X is not divisible by 2.
Since 10^{3n+1} requires a factor of 2^{3n+1}, we conclude that it is not possible that
10^{3n+1}=A^{3}+B^{3} 
Nice Answer, But Please Try To Use (Mod)
That Way Is Easier 
Hey, Your ANSWER is corrupt, cause it doesnt really explain anything! Try to make it more clear.
SY