Triangle Angle Multiples

Let the three angles in the triangle be $a = mx$ , $b = my$ , and $c = mz$ for positive integers $a$ , $b$ , $c$ , $x$ , $y$ , $z$ . Then the angle sum is $a + b + c = 180°$ or $mx + my + mz = 180°$ or $x + y + z = \frac{180°}{m}$ , which is an equation of a plane.

Since $a$ , $b$ , and $c$ are angles in a triangle, we can deduce that $1 \leq x \leq \frac{180°}{m} - 2$ , $1 \leq y \leq \frac{180°}{m} - 2$ , and $1 \leq z \leq \frac{180°}{m} - 2$ , so the number of all integer solutions on the plane within these parameters is

$\displaystyle \sum_{k=1}^{\frac{180}{m} - 2} k = \frac{1}{2}(\frac{180°}{m} - 2)(\frac{180°}{m} - 1) = \frac{16200}{m^2} - \frac{270}{m} + 1$

However, this amount is too much, as it includes all rotations and reflections of scalene triangles $3! = 6$ times (possibilities $abc$ , $acb$ , $bac$ , $bca$ , $cab$ , and $cba$ ), and all rotations and reflections of isosceles triangles (that are not equilateral triangles) $\frac{3!}{2!} = 3$ times (possibilities $aab$ , $aba$ , and $baa$ ).

There is $1$ possible equilateral triangle.

For isosceles triangles (that are not equilateral triangles), the two congruent angles can be any number from $1$ to $\frac{1}{2}(\frac{180°}{m} - 2) = \frac{90°}{m} - 1$ , minus the $1$ equilateral triangle, for $\frac{90°}{m} - 2$ possible isosceles triangles (that are not equilateral triangles).

For scalene triangles, we can take the number of all integer solutions and subtract out $3$ times the number of isosceles triangles (that are not equilateral triangles) and the $1$ equilateral triangle and divide the difference by $6$ . This makes a total of $\frac{1}{6}((\frac{16200}{m^2} - \frac{270}{m} + 1) - 3(\frac{90°}{m} - 2) - 1) = \frac{2700}{m^2} - \frac{90}{m} + 1$ possible scalene triangles.

The total number of possible non-similar triangles is the sum of the possible equilateral, isosceles (that are not equilateral), and scalene triangles, which is $1 + (\frac{90}{m} - 2) + (\frac{2700}{m^2} - \frac{90}{m} + 1) = \frac{2700}{m^2}$ .

In this problem, $\frac{2700}{m^2} = 300$ , which solves to $m = \boxed{3}$ .