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id	title	challengeType	forumTopicId	dashedName
5900f54c1000cf542c51005e	Problem 478: Mixtures	5	302155	problem-478-mixtures

--description--

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: (10 \times \frac{3}{5} + 20 \times \frac{3}{20} + 30 \times \frac{3}{10} : 10 \times \frac{0}{5} + 20 \times \frac{6}{20} + 30 \times \frac{3}{10} : 10 \times \frac{2}{5} + 20 \times \frac{11}{20} + 30 \times \frac{4}{10}) = (18 : 15 : 27) = (6 : 5 : 9)

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:

{(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (0 : 1 : 2), (0 : 2 : 1), (1 : 0 : 0), (1 : 0 : 1), (1 : 0 : 2), (1 : 1 : 0), (1 : 1 : 1), (1 : 1 : 2), (1 : 2 : 0), (1 : 2 : 1), (1 : 2 : 2), (2 : 0 : 1), (2 : 1 : 0), (2 : 1 : 1), (2 : 1 : 2), (2 : 2 : 1)}.

Let E(n) be the number of subsets of M(n) which can produce the mixture with ratio (1 : 1 : 1), i.e., the mixture with equal parts A, B and C.

We can verify that E(1) = 103, E(2) = 520\\,447, E(10)\bmod {11}^8 = 82\\,608\\,406 and E(500)\bmod {11}^8 = 13\\,801\\,403.

Find E(10\\,000\\,000)\bmod {11}^8.

--hints--

mixtures() should return 59510340.

assert.strictEqual(mixtures(), 59510340);

--seed--

--seed-contents--

function mixtures() {

  return true;
}

mixtures();

--solutions--

// solution required

2.3 KiB Raw Blame History