45 lines
1.3 KiB
Markdown
45 lines
1.3 KiB
Markdown
---
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id: 5900f4831000cf542c50ff95
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title: 'Problem 278: Linear Combinations of Semiprimes'
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challengeType: 5
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forumTopicId: 301928
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dashedName: problem-278-linear-combinations-of-semiprimes
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---
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# --description--
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Given the values of integers $1 < a_1 < a_2 < \ldots < a_n$, consider the linear combination $q_1a_1 + q_2a_2 + \ldots + q_na_n = b$, using only integer values $q_k ≥ 0$.
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Note that for a given set of $a_k$, it may be that not all values of $b$ are possible. For instance, if $a_1 = 5$ and $a_2 = 7$, there are no $q_1 ≥ 0$ and $q_2 ≥ 0$ such that $b$ could be 1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18 or 23.
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In fact, 23 is the largest impossible value of $b$ for $a_1 = 5$ and $a_2 = 7$. We therefore call $f(5, 7) = 23$. Similarly, it can be shown that $f(6, 10, 15)=29$ and $f(14, 22, 77) = 195$.
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Find $\sum f(pq,pr,qr)$, where $p$, $q$ and $r$ are prime numbers and $p < q < r < 5000$.
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# --hints--
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`linearCombinationOfSemiprimes()` should return `1228215747273908500`.
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```js
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assert.strictEqual(linearCombinationOfSemiprimes(), 1228215747273908500);
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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 linearCombinationOfSemiprimes() {
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return true;
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}
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linearCombinationOfSemiprimes();
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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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