How do I justify that $\beta_i \leq \alpha_i$ in my attempt at proving the following theorem.
I can neither figure out how to justify the assumption nor how to complete the proof without it.
Theorem:
Let $n$ have prime factorization $p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$.
Then $d$ is a divisor of $n$ if and only if $d$ has prime factorization $p_1^{\beta_1}p_2^{\beta_2}\cdots p_k^{\beta_k}$ with $0 \leq \beta_i \leq \alpha_i$ for all $1\leq i \leq k $.
Proof:
The statement is trivial both ways for $d=1=p_1^{0}p_2^{0}\cdots p_k^{0}$ so throughout we assume that $d>1$.
($\Rightarrow$) Let $n$ have prime factorization $p^{\alpha_1}p^{\alpha_2}\cdots p^{\alpha_k}$ and suppose $d$ is a divisor of $n$. So $n=dd'$ for some $d'>1$.
Consider the prime factorizations of $d$ and $d'$ into primes $\{q_i\}$ and $\{q'_i\}$ respectively ($\{q_i\}$ and $\{q'_i\}$ are not necessarily distinct or disjoint):
$$d=q_1q_2\cdots q_m \quad d=q'_1q'_2\cdots q'_n $$
Since $n=dd'$ by expressing $n$ in terms of its prime fractorization we find that
$$p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}=q_1q_2\cdots q_mq'_1q'_2\cdots q'_n.$$
So by uniqueness of prime factorization each $q_i=p_j$ (Note: there may be many different $q_i$ equal to the same $p_j$), thus collecting like primes together we get find that $d$ has prime factorization
$$d=p_1^{\beta_1}p_2^{\beta_2}\cdots p_k^{\beta_k}.$$
with each $\beta_i \leq \alpha_i$ as else d would not be a divisor of $n$.
Best Answer
Let $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$
And $d = q_1^{\beta_1}q_2^{\beta_2}\cdots q_j^{\alpha_j}$ be unique prime factorizations.
$q_i|d$ so $q_i|n$ so by euclid's lemma $q_i =p_m$ for some $p_m$ and $\{q_j\}\subset \{p_i\}$.
So $d = p_1^{\beta_1}p_2^{\beta_2}\cdots p_k^{\beta_k}$ if we allow some $\beta_i$ to be $0$.
===== important bit below =====
Suppose there is a $\beta_i > \alpha_i$.
Then Let $d'= \frac d{p_i^{\alpha_i}} =p_1^{\beta_1}p_2^{\beta_2}\cdots p_i^{\beta_k - \alpha_k}\cdots p_k^{\beta_k}$ and $n' = \frac n{\alpha_i} = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_i^0 \cdots p_k^{\alpha_k}$
We know: $d'|n'$ that $\beta_i - \alpha_i > 0$. And $p_i \not \mid n'$.
But if $\beta_i - \alpha_i > 0$ then $\beta_i - \alpha_i \ge 1$ and $p_i|d'$ and so $p_i|n'$ which is a contradiction.
So all $\beta_i \le \alpha_i$.