Show that if $a$, $b$ and $c$ are positive integers with $(a,b) = 1$
and $ab = c^n$, then there positive integers $d$ and $e$ such that $a = d^n$ and $b = e^n$.

Question

Show that if $a$, $b$ and $c$ are positive integers with $(a,b) = 1$
and $ab = c^n$, then there positive integers $d$ and $e$ such that $a = d^n$ and $b = e^n$.

anonymous · Answer

write each number in factored form
since the gcd is 1, all the prime factors are different

anonymous · Answer

Could you show me what you mean

anonymous · Answer

$$a=p_1^{\alpha_1}p_2^{\alpha_2}...p_k^{\alpha_k}$$$$b=q_1^{\beta_1}q_2^{\beta_2}...q_k^{\beta_j}$$
where $p_i\neq q_l$

anonymous · Answer

then write $ab=c^n$ and note that since all the prime factors are different , you can write it as $d^n\times e^n$

anonymous · Answer

i guess you should point out that if the product is $c^n$ then all the exponents in each prime factorization is divisible by $n$ so maybe the way i wrote it was not so good

anonymous · Answer

Thanks