Lost on this one

let a, b and c be positive integers such that a^(b+c) = b^(c)c. Prove that b is a divisor of c, and that c is of form d^(b) for some positive integer d.

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- Aug 10th 2012, 02:51 PMfarlapPositive integers
Lost on this one

let a, b and c be positive integers such that a^(b+c) = b^(c)c. Prove that b is a divisor of c, and that c is of form d^(b) for some positive integer d. - Sep 3rd 2012, 04:34 AMwauwauRe: Positive integers
$\displaystyle a^{b+c}=b^cc$

hence $\displaystyle b|a$

or $\displaystyle a=r.b$

for some integer r

$\displaystyle (rb)^{b+c}=b^cc$

$\displaystyle r^{b+c}b^bb^c=b^cc$

$\displaystyle r^{b+c}b^b=c$

hence [Tex]b|c or $\displaystyle c=s.b$ for some integer s

$\displaystyle r^{(1+s)b}b^b = c$

$\displaystyle (r^{1+s}b)^b = c$