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fix(curriculum): clean-up Project Euler 462-480 (#43069)

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* fix: clean-up Project Euler 462-480

* fix: missing image extension

* fix: corrections from review

Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>

Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
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curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-478-mixtures.md
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@ -8,26 +8,32 @@ dashedName: 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.
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: (10·3/5 + 20·3/20 + 30·3/10 : 10·0/5 + 20·6/20 + 30·3/10 : 10·2/5 + 20·11/20 + 30·4/10) = (18 : 15 : 27) = (6 : 5 : 9)
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.
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.
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)}.
For example, $M(2)$ contains the 19 mixtures with the following ratios:
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) = 520447, E(10) mod 118 = 82608406 and E(500) mod 118 = 13801403. Find E(10 000 000) mod 118.
{(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--
`euler478()` should return 59510340.
`mixtures()` should return `59510340`.
```js
assert.strictEqual(euler478(), 59510340);
assert.strictEqual(mixtures(), 59510340);
```
# --seed--
@ -35,12 +41,12 @@ assert.strictEqual(euler478(), 59510340);
## --seed-contents--
```js
function euler478() {
function mixtures() {
return true;
}
euler478();
mixtures();
```
# --solutions--