So I know that I need to form a bijection, I just need the 2 functions for the different sets. I know that the function for all positive integers divisible by 5 is f(x) = 5x, however I have no clue how to find the function that maps the triangle numbers. I've been attempting trial and error, with no success. Is there any process that would make finding the function easier? I've been comparing the differences of each interval, but still nothing. Any hints to the right track would be great.
thanks
Answer
I think that we can just look at how the triangle numbers are constructed and how the positive integers divisible by $5$ are constructed and we can see a bijection naturally.
The $n^{\mathrm{th}}$ triangle number $T_n$ is given as $T_n = 1+2+3+\dotsb+n = \frac{n(n+1)}{2}$. This is one of those series that you will just have to have seen enough times (in a number theory class or discrete math class) in order to recognize consistently.
The $n^{\mathrm{th}}$ multiple of $5$ is given by $5n$.
It looks like it will be easiest to take a bijection from the multiples of $5$ to the triangles numbers, so we can say that we need a function $f$ that does the following:
$$
f(5n) = \frac{n(n+1)}{2}
$$
Having $5n$ as an argument to $f$ is weird so we can just do a little substitution, letting $5n \to x$.
$$
f(5n) = \frac{n(n+1)}{2} \quad\implies\quad
f(x) = \frac{\frac{x}{5}(\frac{x}{5}+1)}{2}
$$
And now $f$ is a function that maps multiples of $5$ to triangle numbers.
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