Let us consider mixtures of three substances: A, B and C. A mixture can be described by a ratio of the amounts of A, B, and C in it, i.e., (a:b:c). For example, a mixture described by the ratio (2:3:5) contains 20% A, 30% B and 50% C.
For the purposes of this problem, we cannot separate the individual components from a mixture. However, we can combine different amounts of different mixtures to form mixtures with new ratios.
For example, say we have three mixtures with ratios (3:0:2), (3:6:11) and (3:3:4). By mixing 10 units of the first, 20 units of the second and 30 units of the third, we get a new mixture with ratio (6:5:9), since:
However, with the same three mixtures, it is impossible to form the ratio (3:2:1), since the amount of B is always less than the amount of C.
Let n be a positive integer. Suppose that for every triple of integers (a, b, c) with 0 ≤ a, b, c ≤ n and gcd(a, b, c) = 1, we have a mixture with ratio (a:b:c). Let M(n) be the set of all such mixtures.
For example, M(2) contains the 19 mixtures with the following ratios: