Counting coprime integers

Non-coprime. Count numbers divisible by $2$, $3$, or $5$:
- $|A_2|=\lfloor60/2\rfloor=30$
- $|A_3|=20$
- $|A_5|=12$
- $|A_2\cap A_3|=|A_6|=10$
- $|A_2\cap A_5|=|A_{10}|=6$
- $|A_3\cap A_5|=|A_{15}|=4$
- $|A_2\cap A_3\cap A_5|=|A_{30}|=2$

Union. $30+20+12-10-6-4+2=44$.

Coprime. $60-44=16$.

Check. $\varphi(60)=60(1-1/2)(1-1/3)(1-1/5)=16$. ✓

Conclusion. $16$ coprime integers.