49 lines
1.4 KiB
Markdown
49 lines
1.4 KiB
Markdown
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---
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id: 5900f4cb1000cf542c50ffdd
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title: 'Problem 350: Constraining the least greatest and the greatest least'
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challengeType: 5
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forumTopicId: 302010
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dashedName: problem-350-constraining-the-least-greatest-and-the-greatest-least
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---
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# --description--
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A list of size $n$ is a sequence of $n$ natural numbers. Examples are (2, 4, 6), (2, 6, 4), (10, 6, 15, 6), and (11).
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The greatest common divisor, or $gcd$, of a list is the largest natural number that divides all entries of the list. Examples: $gcd(2, 6, 4) = 2$, $gcd(10, 6, 15, 6) = 1$ and $gcd(11) = 11$.
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The least common multiple, or $lcm$, of a list is the smallest natural number divisible by each entry of the list. Examples: $lcm(2, 6, 4) = 12$, $lcm(10, 6, 15, 6) = 30$ and $lcm(11) = 11$.
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Let $f(G, L, N)$ be the number of lists of size $N$ with $gcd ≥ G$ and $lcm ≤ L$. For example:
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$$\begin{align} & f(10, 100, 1) = 91 \\\\ & f(10, 100, 2) = 327 \\\\ & f(10, 100, 3) = 1135 \\\\ & f(10, 100, 1000)\bmod {101}^4 = 3\\,286\\,053 \end{align}$$
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Find $f({10}^6, {10}^{12}, {10}^{18})\bmod {101}^4$.
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# --hints--
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`leastGreatestAndGreatestLeast()` should return `84664213`.
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```js
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assert.strictEqual(leastGreatestAndGreatestLeast(), 84664213);
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```
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# --seed--
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## --seed-contents--
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```js
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function leastGreatestAndGreatestLeast() {
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return true;
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}
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leastGreatestAndGreatestLeast();
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```
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# --solutions--
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```js
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// solution required
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```
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