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Setsuna Kujo
Aug 09, 2021

Triangle Numbers

in Math Problems

I started thinking about triangle numbers recently. Specifically, multiples of Triangle numbers. Are there any triangle numbers that when multiplied to a positive integer other than 1 (or 0) give another triangle number? Aside from the obvious fact that 1 is a triangle number and you can multiply that with another triangle to get the same triangle out, what else is there? And what are the properties of these numbers?


Examples off the top of my head:

2*3 = 6

6*6 = 36

12*10 = 120

3*15 = 45

5*21 = 105


Strange.

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Ian Fowler
Aug 09, 2021  ·  Edited: Aug 09, 2021

Infinitely many. t(n) = n(n+1)/2


for example t(6) = (6)(7)/2 = 21 and t(2) = (2)(3)/2 = 3

21 = 7(3)

so t(6) = 7*t(2)


Another one: t(8) = 6*t(3)

Or: t(15) = 12*t(4)


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