0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ The first one as a "rule" : Yes-No. The second one : Yes-No-No. The thirs one : No-Yes-No-No-No-No. $\endgroup$ – Mauro ALLEGRANZA May 17 '18 at 13:51
  • $\begingroup$ 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,...). $\endgroup$ – Mauro ALLEGRANZA May 17 '18 at 13:56
  • 1
    $\begingroup$ @MauroALLEGRANZA: It appears to me they count the first line as $0$, so line $1$ has $5/5$ and does not qualify. $\endgroup$ – Ross Millikan May 17 '18 at 14:16
  • $\begingroup$ 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. $\endgroup$ – DanielWainfleet May 17 '18 at 15:38
3
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.