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| contest_2011-09-27 [2011/09/25 12:42] – jtkorb | contest_2011-09-27 [2011/09/27 11:00] (current) – jtkorb |
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| * D: [[http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=35&page=show_problem&problem=947|Carmichael Numbers]] | * D: [[http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=35&page=show_problem&problem=947|Carmichael Numbers]] |
| * E: [[http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=35&page=show_problem&problem=1045|Euclid Problem]] | * E: [[http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=35&page=show_problem&problem=1045|Euclid Problem]] |
| * F: [[http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=35&page=show_problem&problem=1080|Factovisors]] | * F: [[Prelude to Factovisors]] |
| * G: [[http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=35&page=show_problem&problem=1030|Repackaging]] | * G: [[http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=35&page=show_problem&problem=1080|Factovisors]] |
| | * H: [[http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=35&page=show_problem&problem=1030|Repackaging]] |
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| //Remember:// If you've already solved one or more of these problems, try (1) solving again without referring to your old solution, and/or (2) using a different language (Java or C++). If you want to work on an additional problem from the book, let [[jtk@purdue.edu|me]] know. | //Remember:// If you've already solved one or more of these problems, try (1) solving again without referring to your old solution, and/or (2) using a different language (Java or C++). If you want to work on an additional problem from the book, let [[jtk@purdue.edu|me]] know. |
| * C: Observe that, if ''k'' is even, ''n ^ k mod p'' is ''(n ^ (k/2) mod p) ^ 2 mod p''. Generalize and handle the case for ''k'' odd. | * C: Observe that, if ''k'' is even, ''n ^ k mod p'' is ''(n ^ (k/2) mod p) ^ 2 mod p''. Generalize and handle the case for ''k'' odd. |
| * D: Combine the two preludes. | * D: Combine the two preludes. |
| * E: The GCD algorithm given in the text solves this problem directly. | * E: The Extended Euclid Algorithm for GCD (given in the text), given in the text, solves this problem directly. |
| * F: Theorem: The highest power of ''p'' that divides ''n!'' is ''n/p + n/p^2 + n/p^3 ...''. So, ''p^e'' divides ''n!'' if this sum is at least ''e'' before ''n/p^k'' reaches 0. | * F: Brute force factorization is fine. You'll want to keep a table of ''(p,e)'' pairs for each prime, ''p'', that divides ''e'' times. |
| * G: Use the Diophantine solution method in the text. | * G: Theorem: The highest power of ''p'' that divides ''n!'' is ''n/p + n/p^2 + n/p^3 ...''. So, ''p^e'' divides ''n!'' if this sum is at least ''e'' before ''n/p^k'' reaches 0. |
| | * H: The text suggests using the Diophantine solution method given there. Note that you don't actually need to compute the solution, just determine whether or not a solution exists. |
