My mother and I have been wrecking our brains trying to figure this out.
Q: when is the next time these three fraction won't be whole simultaneously.
Our manual working:
$2/2, 3/3, 4/5$ - start
$1/2, 1/3, 5/5$
$2/2, 2/3, 1/5$
$1/2, 3/3, 2/5$
$2/2, 1/3, 3/5$
$1/2, 2/3, 4/5$ - end/correct
The correct answer is $5$ but we would really appreciate it in an equation. My mother just started working her way through college and I wish I could help her with her homework but I'm only in middle school. Thank you for your time.
It appears you count line numbers from $0$. The first fraction is whole when the line number $n \equiv 0 \pmod 2$. The second fraction is whole when $n \equiv 0 \pmod 3$. The third is whole when $n \equiv 1 \pmod 5$. The Chinese Remainder Theorem tells you that the pattern will repeat every $\operatorname{LCM}(2,3,5)=30$. The first two columns will both not be whole when $n\equiv 1,5 \pmod 6$ and you can see that $n=5$ will have none of them whole. The complete list that have none whole is $$5,7,13,17,19,23,25,29 \pmod{30}$$