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**skoker** Could someone check to see if I got this right. It says 'divisible' not 'evenly divisible'. I think it proves divisibility. And is there a difference between 'divisible by ' and 'factor of'? they seem to be the same thing.

$\displaystyle a^{2n}-b^{2n}$ is divisible by $\displaystyle (a+b)$ if $\displaystyle a \ne -b$ . For all positive integers n.

1. first I prove n=1 is true.

$\displaystyle a^{2(1)}-b^{2(1)} = a^{2}-b^{2} = (a+b)(a-b)$

2. then I assume n=k is true and solve for (k+1)th item.

$\displaystyle a^{2(k+1)}-b^{2(k+1)}$

$\displaystyle a^{2k+2}-b^{2k+2}$

$\displaystyle a^{2k+2}-a^{2k}b^{2}+a^{2k}b^{2}-b^{2k+2}$

$\displaystyle a^{2k}(a^{2}-b^{2})+b^{2}(a^{2k}-b^{2k})$