<s>[INST] Find the Prime Factorization of $\varphi(11!)$ 
Find the Prime Factorization of $\varphi(11!)$

What I did:
$\varphi(11!)=\varphi(11)\cdot\varphi(10)...\varphi(1)$
$$\varphi(11)=2\cdot 5\\
\varphi(10)=2^2\\
\varphi(9)=3\cdot 2\\
\varphi(8)=2^2\\
\varphi(7)=2\cdot 3\\
\varphi(6)=2\\
\varphi(5)=2^2\\
\varphi(4)=2\\
\varphi(3)=2\\
$$
$$\Longrightarrow\text{ the answer is }  2^{12}\cdot 5\cdot 3^2$$
$11!$
$\varphi(11!)=\varphi(39916800)=8294400$
$8294400=2^{12}3^45^2$
Where am I wrong?
Is there a simpler way to solve this?
 [/INST] $\varphi(11!)=(11!) (1-\frac{1}{2})(1-\frac{1}{3}) (1-\frac{1}{5}) (1-\frac{1}{7}) (1-\frac{1}{11})$. I do not really know if this will help.
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