MathLabs

Problem 4

Let ABCABC be an equilateral triangle. From vertex AA we draw a ray toward the interior of the triangle so that it reaches one of the sides. When the ray reaches a side it bounces off following the law of reflection, that is, if it arrives with directed angle θ\theta it leaves with directed angle 180∘−θ180^\circ-\theta. After nn bounces the ray returns to AA without ever landing on either of the other two vertices. Find all possible values of nn.
Step 4 of 7: n is never a multiple of three
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a≡b(mod3), gcd⁡(a,b)=1 ⟹ 3∤na\equiv b\pmod3,\ \gcd(a,b)=1 \ \Longrightarrow\ 3\nmid n
Detailed analysis

If 3∣n=2(a+b)−33\mid n=2(a+b)-3 then 3∣(a+b)3\mid(a+b). Together with a≡b(mod3)a\equiv b\pmod3 this gives 2a≡a+b≡0(mod3)2a\equiv a+b\equiv0\pmod3, so 3∣a3\mid a, and then 3∣b3\mid b as well; but then 33 divides gcd⁡(a,b)=1\gcd(a,b)=1, a contradiction. Hence 3∤n3\nmid n.