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14310 - Revision history
2026-05-02T01:48:55Z
Revision history for this page on the wiki
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https://wiki.universitas.ro/index.php?title=14310&diff=10561&oldid=prev
Andrei.Horvat at 18:24, 12 January 2025
2025-01-12T18:24:29Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:24, 12 January 2025</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie <math>n, n + 2, n + 6 </math> trei numere naturale și <math> S </math> suma lor.''</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie <math>n, n + 2, n + 6 </math> trei numere naturale și <math> S </math> suma lor.''</div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=14310&diff=10558&oldid=prev
Benzar Ioan at 17:58, 12 January 2025
2025-01-12T17:58:30Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:58, 12 January 2025</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie <math>n, n + 2, n + 6 </math> trei numere naturale și <math> S </math> suma lor.''</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie <math>n, n + 2, n + 6 </math> trei numere naturale și <math> S </math> suma lor.''</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>a) ''Dați exemplu de cel puțin trei valori pentru <math> n \in \mathbb{N}</math> astfel <del style="font-weight: bold; text-decoration: none;">incat </del>numerele <math> n, n + 2, n + 6 </math> <del style="font-weight: bold; text-decoration: none;">sa </del>fie simultan numere prime.'' <p></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>a) ''Dați exemplu de cel puțin trei valori pentru <math> n \in \mathbb{N}</math> astfel <ins style="font-weight: bold; text-decoration: none;">încât </ins>numerele <math> n, n + 2, n + 6 </math> <ins style="font-weight: bold; text-decoration: none;">să </ins>fie simultan numere prime.'' <p></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>b) ''<del style="font-weight: bold; text-decoration: none;">Daca </del><math> n, n + 2, n + 6 </math> sunt simultan numere prime, <del style="font-weight: bold; text-decoration: none;">aratati ca exista </del><math> k \in \mathbb{N} </math> astfel <del style="font-weight: bold; text-decoration: none;">incat </del><math> S = 9k + 5</math>. '' <p></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>b) ''<ins style="font-weight: bold; text-decoration: none;">Dacă </ins><math> n, n + 2, n + 6 </math> sunt simultan numere prime, <ins style="font-weight: bold; text-decoration: none;">arătați că există </ins><math> k \in \mathbb{N} </math> astfel <ins style="font-weight: bold; text-decoration: none;">încât </ins><math> S = 9k + 5</math>. '' <p></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>c) ''<del style="font-weight: bold; text-decoration: none;">Daca </del><math> n, n + 2, n + 6 </math> sunt numere prime, <del style="font-weight: bold; text-decoration: none;">determinati </del>restul <del style="font-weight: bold; text-decoration: none;">impartirii numarului </del><math> S</math> la <math> 18 </math>.''</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>c) ''<ins style="font-weight: bold; text-decoration: none;">Dacă </ins><math> n, n + 2, n + 6 </math> sunt numere prime, <ins style="font-weight: bold; text-decoration: none;">determinați </ins>restul <ins style="font-weight: bold; text-decoration: none;">împărțirii numărului </ins><math> S</math> la <math> 18 </math>.''</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''<del style="font-weight: bold; text-decoration: none;">Solutie</del>:''' <p></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''<ins style="font-weight: bold; text-decoration: none;">Soluție</ins>:''' <p></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>a) Pentru <math> n = 5 </math> numerele sunt <math> 5, 7, 11 </math>. Pentru <math> n = 11 </math> avem <math> 11, 13, 17 </math>, iar pentru <math> n = 17 </math> <del style="font-weight: bold; text-decoration: none;">obtinem </del><math> 17, 19, 23 </math>. <p></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>a) Pentru <math> n = 5 </math> numerele sunt <math> 5, 7, 11 </math>. Pentru <math> n = 11 </math> avem <math> 11, 13, 17 </math>, iar pentru <math> n = 17 </math> <ins style="font-weight: bold; text-decoration: none;">obținem </ins><math> 17, 19, 23 </math>. <p></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>b) Se <del style="font-weight: bold; text-decoration: none;">stie ca </del>numerele prime au forma <math> 6p + 1 </math> sau <math> 6p + 5 </math>. <del style="font-weight: bold; text-decoration: none;">Daca </del><math> n = 6p + 1</math>, atunci <math> n + 2 = 6p + 3 </math> care nu este numar prim. <del style="font-weight: bold; text-decoration: none;">Asadar</del>, <math> n = 6p + 5 </math>. <del style="font-weight: bold; text-decoration: none;">In aceasta situatie </del>avem <math> S = 6p + 5 + 6p + 7 + 6p + 11 = 18p + 23 = 9(2p + 2) + 5 </math>. <del style="font-weight: bold; text-decoration: none;">In </del>concluzie, <del style="font-weight: bold; text-decoration: none;">exista </del><math> k = 2p + 2 </math> astfel <del style="font-weight: bold; text-decoration: none;">incat </del><math> S = 9k + 5 </math>. <p></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>b) Se <ins style="font-weight: bold; text-decoration: none;">știe că </ins>numerele prime au forma <math> 6p + 1 </math> sau <math> 6p + 5 </math>. <ins style="font-weight: bold; text-decoration: none;">Dacă </ins><math> n = 6p + 1</math>, atunci <math> n + 2 = 6p + 3 </math> care nu este numar prim. <ins style="font-weight: bold; text-decoration: none;">Așadar</ins>, <math> n = 6p + 5 </math>. <ins style="font-weight: bold; text-decoration: none;">În această situație </ins>avem <math> S = 6p + 5 + 6p + 7 + 6p + 11 = 18p + 23 = 9(2p + 2) + 5 </math>. <ins style="font-weight: bold; text-decoration: none;">În </ins>concluzie, <ins style="font-weight: bold; text-decoration: none;">există </ins><math> k = 2p + 2 </math> astfel <ins style="font-weight: bold; text-decoration: none;">încât </ins><math> S = 9k + 5 </math>. <p></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>c) Din punctul b) avem <math> S = 18(p + 1) + 5 </math>, deci restul <del style="font-weight: bold; text-decoration: none;">impartirii </del>lui <math> S </math> la <math> 18 </math> este <math> 5 </math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>c) Din punctul b) avem <math> S = 18(p + 1) + 5 </math>, deci restul <ins style="font-weight: bold; text-decoration: none;">împărțirii </ins>lui <math> S </math> la <math> 18 </math> este <math> 5 </math>.</div></td></tr>
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Benzar Ioan
https://wiki.universitas.ro/index.php?title=14310&diff=10556&oldid=prev
Benzar Ioan: Created page with "'''14310 (Traian Covaciu)''' ''Fie <math>n, n + 2, n + 6 </math> trei numere naturale și <math> S </math> suma lor.'' a) ''Dați exemplu de cel puțin trei valori pentru <math> n \in \mathbb{N}</math> astfel incat numerele <math> n, n + 2, n + 6 </math> sa fie simultan numere prime.'' <p> b) ''Daca <math> n, n + 2, n + 6 </math> sunt simultan numere prime, aratati ca exista <math> k \in \mathbb{N} </math> astfel incat <math> S = 9k + 5</math>. '' <p> c) ''Daca <math..."
2025-01-12T17:50:29Z
<p>Created page with "'''14310 (Traian Covaciu)''' ''Fie <math>n, n + 2, n + 6 </math> trei numere naturale și <math> S </math> suma lor.'' a) ''Dați exemplu de cel puțin trei valori pentru <math> n \in \mathbb{N}</math> astfel incat numerele <math> n, n + 2, n + 6 </math> sa fie simultan numere prime.'' <p> b) ''Daca <math> n, n + 2, n + 6 </math> sunt simultan numere prime, aratati ca exista <math> k \in \mathbb{N} </math> astfel incat <math> S = 9k + 5</math>. '' <p> c) ''Daca <math..."</p>
<p><b>New page</b></p><div>'''14310 (Traian Covaciu)'''<br />
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''Fie <math>n, n + 2, n + 6 </math> trei numere naturale și <math> S </math> suma lor.''<br />
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a) ''Dați exemplu de cel puțin trei valori pentru <math> n \in \mathbb{N}</math> astfel incat numerele <math> n, n + 2, n + 6 </math> sa fie simultan numere prime.'' <p><br />
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b) ''Daca <math> n, n + 2, n + 6 </math> sunt simultan numere prime, aratati ca exista <math> k \in \mathbb{N} </math> astfel incat <math> S = 9k + 5</math>. '' <p><br />
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c) ''Daca <math> n, n + 2, n + 6 </math> sunt numere prime, determinati restul impartirii numarului <math> S</math> la <math> 18 </math>.''<br />
<p><br />
'''Solutie:''' <p><br />
a) Pentru <math> n = 5 </math> numerele sunt <math> 5, 7, 11 </math>. Pentru <math> n = 11 </math> avem <math> 11, 13, 17 </math>, iar pentru <math> n = 17 </math> obtinem <math> 17, 19, 23 </math>. <p><br />
b) Se stie ca numerele prime au forma <math> 6p + 1 </math> sau <math> 6p + 5 </math>. Daca <math> n = 6p + 1</math>, atunci <math> n + 2 = 6p + 3 </math> care nu este numar prim. Asadar, <math> n = 6p + 5 </math>. In aceasta situatie avem <math> S = 6p + 5 + 6p + 7 + 6p + 11 = 18p + 23 = 9(2p + 2) + 5 </math>. In concluzie, exista <math> k = 2p + 2 </math> astfel incat <math> S = 9k + 5 </math>. <p><br />
c) Din punctul b) avem <math> S = 18(p + 1) + 5 </math>, deci restul impartirii lui <math> S </math> la <math> 18 </math> este <math> 5 </math>.</div>
Benzar Ioan