Okay, let's try to figure out this problem. So, it's about a regular polygon where the interior angle is 6.5 times the exterior angle. I need to find the number of sides. Hmm, regular polygons have all sides and angles equal, right? So, both the interior and exterior angles are the same for each vertex. 

First, I remember that for any polygon, the sum of the interior and exterior angles at each vertex is 180 degrees because they are supplementary. That makes sense since they form a straight line. So, if the interior angle is 6.5 times the exterior angle, maybe I can set up an equation using that relationship. Let me write that down.

Let's denote the measure of the exterior angle as E. Then, the interior angle would be 6.5E. Since they are supplementary, E + 6.5E = 180 degrees. Combining like terms, that's 7.5E = 180. Then, solving for E, I divide both sides by 7.5. Let me calculate that. 180 divided by 7.5... Hmm, 7.5 times 24 is 180 because 7.5 times 20 is 150, and 7.5 times 4 is 30, so 150 + 30 = 180. So, E = 24 degrees. 

So, the measure of each exterior angle is 24 degrees. Now, I need to relate this to the number of sides. I recall that the sum of all exterior angles of any polygon is 360 degrees. Since it's a regular polygon, each exterior angle is equal, so if there are n sides, each exterior angle is 360/n degrees. 

Therefore, 360/n = 24. Solving for n, I multiply both sides by n: 360 = 24n. Then, divide both sides by 24: n = 360/24. Let me compute that. 24 times 15 is 360 because 24*10=240 and 24*5=120, so 240+120=360. So, n=15. 

Wait, so the polygon has 15 sides? Let me double-check. If each exterior angle is 24 degrees, then 360 divided by 24 is indeed 15. And the interior angle would be 180 - 24 = 156 degrees. Then, 156 divided by 24 is 6.5, which matches the given condition. So, that seems correct. 

I think that's the answer. Let me just recap to make sure I didn't skip any steps. The key was setting up the equation with the exterior angle and using the fact that the sum of exterior angles is 360. Then solving for n. Yep, that all checks out. No mistakes here.
Therefore, the regular polygon has $\boxed{15}$ sides.
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