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15347 - Revision history
2026-05-02T01:21:34Z
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https://wiki.universitas.ro/index.php?title=15347&diff=10481&oldid=prev
Andrei.Horvat at 08:01, 3 January 2025
2025-01-03T08:01:21Z
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<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>'''15347 (Meda Bojor, Baia Mare)'''</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;">E:</ins>15347 (Meda Bojor, Baia Mare)'''</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"></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>''Determinați numerele prime <math>p, q, r, s</math> pentru care este adevărată relația <math>p \cdot q \cdot r \cdot s = 26(p + q + r + s)</math>.''</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>''Determinați numerele prime <math>p, q, r, s</math> pentru care este adevărată relația <math>p \cdot q \cdot r \cdot s = 26(p + q + r + s)</math>.''</div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=15347&diff=10478&oldid=prev
Zmicala Narcis at 17:01, 2 January 2025
2025-01-02T17:01:06Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:01, 2 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>'''15347 (Meda Bojor, Baia Mare)'''</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>'''15347 (Meda Bojor, Baia Mare)'''</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;"><br /></del></div></td><td colspan="2" class="diff-side-added"></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;"><br /></del></div></td><td colspan="2" class="diff-side-added"></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;">''Determinați numerele prime p, q, r, s pentru care este adevărată relația p · q · r · s = 26(p + q + r + s).'' </del></div></td><td colspan="2" class="diff-side-added"></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;"><br /></del></div></td><td colspan="2" class="diff-side-added"></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;"><br /></del></div></td><td colspan="2" class="diff-side-added"></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;">'''Soluție:''' </del></div></td><td colspan="2" class="diff-side-added"></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;"><br /></del></div></td><td colspan="2" class="diff-side-added"></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;">Deoarece 26 | p · q · r · s și p, q, r, s sunt numere prime, deducem că unul dintre numere este 2, iar altul este 13. Fie p = 2 și q = 13. Atunci relația devine: </del></div></td><td colspan="2" class="diff-side-added"></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;">r · s = r + s + 15, </del></div></td><td colspan="2" class="diff-side-added"></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;">sau </del></div></td><td colspan="2" class="diff-side-added"></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;">r · s - r - s = 16, </del></div></td><td colspan="2" class="diff-side-added"></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;">pe care o putem scrie: </del></div></td><td colspan="2" class="diff-side-added"></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;">(r - 1)(s - 1) = 16. </del></div></td><td colspan="2" class="diff-side-added"></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>Cum r și s sunt numere prime, <del style="font-weight: bold; text-decoration: none;">găseam </del>r = s = 5. Dacă <del style="font-weight: bold; text-decoration: none;">alegem </del>p = 2 și r = 13, găseam q = s = 5, și așa mai departe; toate combinațiile posibile.</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;">''Determinați numerele prime <math>p, q, r, s</math> pentru care este adevărată relația <math>p \cdot q \cdot r \cdot s = 26(p + q + r + s)</math>.''</ins></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;"><br /></del></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> </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;"><br /></del></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></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></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> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></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;">Deoarece <math>26 \mid p \cdot q \cdot r \cdot s</math> și <math>p, q, r, s</math> sunt numere prime, deducem că unul dintre numere este <math>2</math>, iar altul este <math>13</math>. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></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> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></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;">Fie <math>p = 2</math> și <math>q = 13</math>. Atunci relația devine <math>r \cdot s = r + s + 15</math> sau <math>r \cdot s - r - s + 1 = 16</math>, pe care o putem scrie: <math>(r - 1)(s - 1) = 16</math>.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></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> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></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>Cum <ins style="font-weight: bold; text-decoration: none;"><math></ins>r<ins style="font-weight: bold; text-decoration: none;"></math> </ins>și <ins style="font-weight: bold; text-decoration: none;"><math></ins>s<ins style="font-weight: bold; text-decoration: none;"></math> </ins>sunt numere prime, <ins style="font-weight: bold; text-decoration: none;">găsim <math></ins>r = s = 5<ins style="font-weight: bold; text-decoration: none;"></math></ins>. Dacă <ins style="font-weight: bold; text-decoration: none;">alegeam <math></ins>p = 2<ins style="font-weight: bold; text-decoration: none;"></math> </ins>și <ins style="font-weight: bold; text-decoration: none;"><math></ins>r = 13<ins style="font-weight: bold; text-decoration: none;"></math></ins>, găseam <ins style="font-weight: bold; text-decoration: none;"><math></ins>q = s = 5<ins style="font-weight: bold; text-decoration: none;"></math></ins>, și așa mai departe; toate combinațiile posibile.</div></td></tr>
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Zmicala Narcis
https://wiki.universitas.ro/index.php?title=15347&diff=10476&oldid=prev
Zmicala Narcis at 16:09, 2 January 2025
2025-01-02T16:09:28Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:09, 2 January 2025</td>
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Zmicala Narcis
https://wiki.universitas.ro/index.php?title=15347&diff=10475&oldid=prev
Zmicala Narcis: Created page with "'''E:15347 (Meda Bojor, Baia Mare)''' <br /> <br /> ''Determinați numerele prime p, q, r, s pentru care este adevărată relația p · q · r · s = 26(p + q + r + s).'' <br /> <br /> '''Soluție:''' <br /> Deoarece 26 | p · q · r · s și p, q, r, s sunt numere prime, deducem că unul dintre numere este 2, iar altul este 13. Fie p = 2 și q = 13. Atunci relația devine: r · s = r + s + 15, sau r · s - r - s = 16, pe care o putem scrie: (r - 1)(s - 1)..."
2025-01-02T16:07:56Z
<p>Created page with "'''E:15347 (Meda Bojor, Baia Mare)''' <br /> <br /> ''Determinați numerele prime p, q, r, s pentru care este adevărată relația p · q · r · s = 26(p + q + r + s).'' <br /> <br /> '''Soluție:''' <br /> Deoarece 26 | p · q · r · s și p, q, r, s sunt numere prime, deducem că unul dintre numere este 2, iar altul este 13. Fie p = 2 și q = 13. Atunci relația devine: r · s = r + s + 15, sau r · s - r - s = 16, pe care o putem scrie: (r - 1)(s - 1)..."</p>
<p><b>New page</b></p><div>'''E:15347 (Meda Bojor, Baia Mare)'''<br />
<br /><br />
<br /><br />
''Determinați numerele prime p, q, r, s pentru care este adevărată relația p · q · r · s = 26(p + q + r + s).'' <br />
<br /><br />
<br /><br />
'''Soluție:''' <br />
<br /><br />
Deoarece 26 | p · q · r · s și p, q, r, s sunt numere prime, deducem că unul dintre numere este 2, iar altul este 13. Fie p = 2 și q = 13. Atunci relația devine: <br />
r · s = r + s + 15, <br />
sau <br />
r · s - r - s = 16, <br />
pe care o putem scrie: <br />
(r - 1)(s - 1) = 16. <br />
<br />
Cum r și s sunt numere prime, găseam r = s = 5. Dacă alegem p = 2 și r = 13, găseam q = s = 5, și așa mai departe; toate combinațiile posibile.<br />
<br /><br />
<br /></div>
Zmicala Narcis