# Find an integer n such that $U(n)$ is isomorphic to $\Bbb Z_2⊕\Bbb Z_4⊕\Bbb Z_9$

Find an integer n such that $U(n)$ is isomorphic to $Z_2⊕Z_4⊕Z_9$

I have gotten this far: I know $Z_2$ is isomorphic to $U(4)$ and $Z_4$ is isomorphic to $U(5)$. However, I'm having trouble figuring out what $Z_9$ is isomorphic to in regards to the U-group. I remember proving somewhere that for all integers $n\geq 3$, $|U(n)|$ is even. Since $|Z_9|=9$, which is odd, I can't see to find a $U$ group that $Z_9$ is isomorphic to.

Thanks for all the help!

• it is known that $|U(n)|$ is the value Eulerphi(n), but Eulerphi(n) is always even Commented Nov 13, 2015 at 0:10
• Yes, I know - that is where I am running into trouble. Since I can't find a U group that is isomorphic, I am not quite sure how to proceed with this problem. Commented Nov 13, 2015 at 0:15
• the good news is that $73$ is prime : D Commented Nov 13, 2015 at 0:15
• @janmarqz $U_{73}$ is the wrong answer, though, since it is cyclic. Commented Nov 13, 2015 at 0:19
• Plenty of candidates - you just need the nine to combine with either $4$ or $2$, which both can do. The smallest $5\cdot 3^3$ has the smallest prime divisors. Commented Nov 13, 2015 at 0:30

$\mathbb{Z}_2\oplus \mathbb{Z}_4 \oplus \mathbb{Z}_9\cong\mathbb{Z}_2 \oplus \mathbb{Z}_{36}.$
$2+1$ and $36+1$ are distinct primes, so $\mathbb{Z}_3\oplus \mathbb{Z}_{37}\cong \mathbb{Z}_{3.37}$ would work.
Similarly $\mathbb{Z}_2\oplus \mathbb{Z}_4\oplus \mathbb{Z}_9\cong \mathbb{Z}_4\oplus \mathbb{Z}_{18}$.
Then $4+1$ and $18+1$ are also distinct primes, so $\mathbb{Z}_5\oplus \mathbb{Z}_{19}\cong \mathbb{Z}_{5.19}$ would also work.
• So the better problem would be find all $n$ instead of one. Commented Nov 13, 2015 at 4:53