Q1: Two plants, a rose and a jasmine, grow up and around a cylindrical tree trunk. They start from the same point at the foot of the tree, but the rose goes clockwise and the jasmine counterclockwise around the trunk. When the two plants meet at the first branch, the rose has made three circles around the trunk and the jasmine has made five circles. How many times did the plants cross between the foot of the tree and the first branch?
2 Answers
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Pardon the ugly paintjob, but perhaps it is what you are looking for:
Consider the following image:
I'm hoping it is clear from this image that this problem is equivalent so solving the equation
$$ \frac{x}{5} = -\frac{x}{3} \pmod{1/5},\qquad 0\leq x< 3$$ [I should perhaps explain that modulo a fraction means the same thing as mod an integer: $x\equiv y\pmod r \iff (x-y)=\ell r$ for some natural $\ell$. Here I'm considering solutions in the real interval $[0,3)$ only.]
In other words, $$ \frac{8x}{3} = 0 \pmod 1,\qquad 0\leq x < 3,$$ substituting $x = 3x'$, we want to solve $$ 8x' = 0 \pmod 1,\qquad 0\leq x' < 1,$$ an equation with solutions $$ x' \in\{ 0, \frac18, \frac28, \dots, \frac78\}.$$
That explains why there are 8 solutions. (Or 7 or 9 depending on whether or not you include the meeting points of these plants at the foot of the tree and at the first branch.)
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From what I understand, x in the answer above means where the two plants cross each other. I thought that since no where did it mention the plants grew evenly, the answer could vary. I am a bit confused right now.
