Question 76
We call a positive integer *n lucky* if it has at least two prime divisors and can be written in the form
$$
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
$$
where $p_1, \dots, p_k$ are distinct prime divisors of $n$. (Note that $n$ may have other prime divisors than $p_1, \dots, p_k$.) Prove that for every prime number $p$, there exists a lucky number $n$ such that $p \mid n$.
$$
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
$$
where $p_1, \dots, p_k$ are distinct prime divisors of $n$. (Note that $n$ may have other prime divisors than $p_1, \dots, p_k$.) Prove that for every prime number $p$, there exists a lucky number $n$ such that $p \mid n$.