Question in Fractions: Assume we have a stick divided into 7 equal parts....

[Nick S.]

Assume we have a stick. We divide it into 7 equal parts, and mark the divison points in a red marker. We also divide it into 13 equal parts, and mark the division points in a green marker. Then we cut the stick into 20 equal parts. Prove that on every part (excluding the first and the last) we have only one divison line - red or green.

This is the question I was asked to prove in class, it is really hard, and I don't know how to approach it, I would be very happy for some help on how can I start solving it...
I need to prove it using words, without drawings, so drawing the stick in not a viable solution.

Thanks everyone who helps

 

[ksdhart2]

Well, perhaps working with an example with actual concrete numbers will help you to see what's going on here. To make the divisions easier, let's say your stick is 1820 units long (chosen because this the least common multiple of 7, 13, and 20). If you divide the stick into 7 equal parts and draw red lines at the division points, where will the red lines be drawn? If you divide the stick into 13 equal parts and draw green lines at the division points, where will the green lines be drawn? As a hint, think about how long, in units, each of the 7 (or 13) segments would be.

Now, imagine cutting the stick into 20 equal parts. You'll end up with 20 segments. How long will each segment be? The problem asks you to prove that every segment, except the first and the last, has only one line. So, where does the second segment start? Where does it end? Is there a line in that segment? What about the third segment? The fourth? Are you noticing a pattern?

As a further exercise to see if you're really understanding what's happening, to be sure you're actually proving what you're meant to be proving, consider what would be happen if the numbers in the problem were changed. That is, consider the case where you divide your stick into 6 parts and draw red lines, then 8 parts and draw green lines. Then cut it into 14 parts. Are there any segments that have multiple lines? If there are, why do you suppose there are in this case, but not in the 7/13/20 case? If there aren't, are there any similarities between these two cases that would suggest why each segment can only have one line?

 

[Steven G]

Assume we have a stick. We divide it into 7 equal parts, and mark the divison points in a red marker. We also divide it into 13 equal parts, and mark the division points in a green marker. Then we cut the stick into 20 equal parts. Prove that on every part (excluding the first and the last) we have only one divison line - red or green.

This is the question I was asked to prove in class, it is really hard, and I don't know how to approach it, I would be very happy for some help on how can I start solving it...
I need to prove it using words, without drawings, so drawing the stick in not a viable solution.

Thanks everyone who helps

lcm(7, 13, 20) = 1820
Suppose the length is 1820 Units long

Now for 7. We have a mark at multiple of 13*20=260-----260, 520, 780,...,1830

For 12 we have a mark at multiple of 140----140, 280, 420,...,1280

For 20 we have marks at multiple of 91----91, 182, 273,...,1820

Can you finish from here????

 

[Nick S.]

lcm(7, 13, 20) = 1820
Suppose the length is 1820 Units long

Now for 7. We have a mark at multiple of 13*20=260-----260, 520, 780,...,1830

For 12 we have a mark at multiple of 140----140, 280, 420,...,1280

For 20 we have marks at multiple of 91----91, 182, 273,...,1820

Can you finish from here????

Yeah sure. Thanks a lot!