# Positive integers

• Aug 10th 2012, 03:51 PM
farlap
Positive 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, 05:34 AM
wauwau
Re: Positive integers
$a^{b+c}=b^cc$
hence $b|a$
or $a=r.b$
for some integer r
$(rb)^{b+c}=b^cc$

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

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

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

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

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