An Isomorphism From $U(st)$ To $U(s)\oplus U(t)$ When $s$ And $t$ Are Relatively Prime

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Problem 105 (An Isomorphism From $U(st)$ To $U(s)\oplus U(t)$ When $s$ And $t$ Are Relatively Prime)

Recall that $\varphi$, the Euler-phi function is defined by $\varphi(n)=|U(n)|$, the order of $U(n)$. Suppose $n=st$ where $s$ and $t$ are relatively prime. In this problem you'll show that $\varphi(st)=\varphi(s)\varphi(t)$.

1. Show that $f:U(st)\to U(s)\oplus U(t)$ defined by $f(x)=(x\mod s,x\mod t)$ is an isomorphism.
2. Why does $\varphi(st)=\varphi(s)\varphi(t)$ when $s$ and $t$ are relatively prime?
3. Use the results above to compute $\varphi(17\cdot 19)$ and $\varphi(15\cdot 63)$.

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