Question 180
Let $a_{0}, a_{1}, \ldots, a_{d}$ be integers such that $\operatorname{GCD}(a_{0}, a_{1})=1$. For every integer $n \geqslant 1$, set
$$
u_{n}=\sum_{k=0}^{d} a_{k} \varphi(n+k)$$
Prove that 1 is the only natural number dividing all the integers $u_{n}$.
Recall that $\varphi(n+k)$ is the number of natural numbers $\ell \lt n+k$ such that $\operatorname{GCD}(\ell, n+k)=1$.
$$
u_{n}=\sum_{k=0}^{d} a_{k} \varphi(n+k)$$
Prove that 1 is the only natural number dividing all the integers $u_{n}$.
Recall that $\varphi(n+k)$ is the number of natural numbers $\ell \lt n+k$ such that $\operatorname{GCD}(\ell, n+k)=1$.