# Fractions Help.

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.

• The first one as a "rule" : Yes-No. The second one : Yes-No-No. The thirs one : No-Yes-No-No-No-No. – Mauro ALLEGRANZA May 17 '18 at 13:51
• So starting with $1/2, 1/3, 5/5$ as line N.1, the line where the three fractions will meet will be : odd, not a multiple of three and not one-more a multiple of $5$ (i.e. not 1,6,11,...). – Mauro ALLEGRANZA May 17 '18 at 13:56
• @MauroALLEGRANZA: It appears to me they count the first line as $0$, so line $1$ has $5/5$ and does not qualify. – Ross Millikan May 17 '18 at 14:16
• I find this completely incomprehensible. After the 3rd line you present some lines of numbers with no statements of what is going on,... no words at all. – DanielWainfleet May 17 '18 at 15:38

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 $$5,7,13,17,19,23,25,29 \pmod{30}$$