Suppose there exists an integer $n$ such that $\varphi(n)=n/4$. Then $n$ must be a multiple of $4$ because $\varphi(n)$ is an integer. Let the prime factorization of $n$ be $$n=2^mp_1^{k_1}p_2^{k_2}\cdots p_j^{k_j}$$ where $m\ge 2$. Now $$\varphi(n)=2^{m-1}p_1^{k_1-1}p_2^{k_2-1}\cdots p_j^{k_j-1}(p_1-1)(p_2-1)\cdots(p_j-1)$$

Setting $\varphi(n)=n/4$ and canceling common terms will yield $$p_1p_2\cdots p_j = 2(p_1-1)(p_2-1)\cdots(p_j-1)$$

But this relation cannot hold because the left side is odd and the right side is even.