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Ligne 75 :
:<math>x = 0 \ \ \mbox{(mod 11)}</math>
 
4.
4.<table border="1">
{| border="1"
<tr>
|-----
<td>0</td>
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7
<td>1</td>
| 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17
<td>2</td>
| 18 ||
<td>3</td>
|-----
<td>4</td>
| || 1 || || || || ||
<td>5</td>
||
<td>6</td>
| || || || || ||
<td>7</td>
| || || || || || mod 2
<td>8</td>
|-----
<td>9</td>
| || 1 || 2 || ||
<td>10</td>
| || || || ||
<td>11</td>
|| ||
<td>12</td>
| || || ||
<td>13</td>
||
<td>14</td>
| || || mod 3
<td>15</td>
|-----
<td>16</td>
| || 1 || || 3 || || ||
<td>17</td>
||
<td>18</td>
| || || || || ||
<td></td>
| || || || || || mod 4
</tr>
|-----
<tr>
| || 1 || 3 || 2 || 4 || || ||
<td></td>
| || || ||
<td>1</td>
| || || ||
<td></td>
||
<td></td>
| || || mod 5
<td></td>
|-----
<td></td>
| || 1 || || || || 5 ||
<td></td>
||
<td></td>
| || || || || ||
<td></td>
| || || || || || mod 6
<td></td>
|-----
<td></td>
| || 1 || 4 || 5 || 2 || 3 || 6 || || ||
<td></td>
|| ||
<td></td>
| || || ||
<td></td>
||
<td></td>
| || || mod 7
<td></td>
|-----
<td></td>
| || 1 || || 3 || || 5 ||
<td></td>
|| 7
<td></td>
| || || || || ||
<td>mod 2</td>
| || || || || || mod 8
</tr>
|-----
<tr>
| || 1 || 5 || || 7
<td></td>
| 2 || || 4
<td>1</td>
| 8 || || ||
<td>2</td>
| || || ||
<td></td>
||
<td></td>
| || || mod 9
<td></td>
|-----
<td></td>
| || 1 || || 7 || || ||
<td></td>
|| 3
<td></td>
| || 9 || || || ||
<td></td>
| || || || || || mod 10
<td></td>
|-----
<td></td>
| || 1 || 6 || 4 || 3 || 9 || 2 || 8 || 7 || 5 || 10 ||
<td></td>
| || || ||
<td></td>
||
<td></td>
| || || mod 11
<td></td>
|-----
<td></td>
| || 1 || || || || 5 ||
<td></td>
|| 7
<td></td>
| || || || 11 || ||
<td>mod 3</td>
| || || || || || mod 12
</tr>
|-----
<tr>
| || 1 || 7 || 9 || 10 || 8 || 11 || 2
<td></td>
| 5 || 3 || 4 || 6 || 12 || || ||
<td>1</td>
||
<td></td>
| || || mod 13
<td>3</td>
|-----
<td></td>
| || 1 || || 5 || || 3 ||
<td></td>
||
<td></td>
| || 11 || || 9 || || 13
<td></td>
| || || || || || mod 14
<td></td>
|-----
<td></td>
| || 1 || 8 || || 4
<td></td>
| || || 13 || 2 ||
<td></td>
|| || 11
<td></td>
| || 7 || 14 ||
<td></td>
||
<td></td>
| || || mod 15
<td></td>
|-----
<td></td>
| || 1 || || 11 || || 13 ||
<td></td>
|| 7
<td></td>
| || 9 || || 3 || || 5
<td>mod 4</td>
| || 15 || || || || mod 16
</tr>
|-----
<tr>
| || 1 || 9 || 6 || 13 || 7 || 3 || 5
<td></td>
| 15 || 2 || 12 || 14 || 10 || 4 || 11 || 8 || 16 ||
<td>1</td>
| || mod 17
<td>3</td>
|-----
<td>2</td>
| || 1 || || || || 11 ||
<td>4</td>
|| 13
<td></td>
| || || || 5 || || 7
<td></td>
| || || || 17 || || mod 18
<td></td>
|-----
<td></td>
| || 1 || 10 || 13 || 5 || 4 || 16 || 11
<td></td>
| 12 || 17 || 2 || 7 || 8 || 3 || 15 || 14 || 6 || 9
<td></td>
| 18 || mod 19
<td></td>
|}
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 5</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td></td>
<td></td>
<td></td>
<td>5</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 6</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td>4</td>
<td>5</td>
<td>2</td>
<td>3</td>
<td>6</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 7</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td></td>
<td>3</td>
<td></td>
<td>5</td>
<td></td>
<td>7</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 8</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td>5</td>
<td></td>
<td>7</td>
<td>2</td>
<td></td>
<td>4</td>
<td>8</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 9</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td></td>
<td>7</td>
<td></td>
<td></td>
<td></td>
<td>3</td>
<td></td>
<td>9</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 10</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td>6</td>
<td>4</td>
<td>3</td>
<td>9</td>
<td>2</td>
<td>8</td>
<td>7</td>
<td>5</td>
<td>10</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 11</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td></td>
<td></td>
<td></td>
<td>5</td>
<td></td>
<td>7</td>
<td></td>
<td></td>
<td></td>
<td>11</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 12</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td>7</td>
<td>9</td>
<td>10</td>
<td>8</td>
<td>11</td>
<td>2</td>
<td>5</td>
<td>3</td>
<td>4</td>
<td>6</td>
<td>12</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 13</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td></td>
<td>5</td>
<td></td>
<td>3</td>
<td></td>
<td></td>
<td></td>
<td>11</td>
<td></td>
<td>9</td>
<td></td>
<td>13</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 14</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td>8</td>
<td></td>
<td>4</td>
<td></td>
<td></td>
<td>13</td>
<td>2</td>
<td></td>
<td></td>
<td>11</td>
<td></td>
<td>7</td>
<td>14</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>mod 15</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td></td>
<td>11</td>
<td></td>
<td>13</td>
<td></td>
<td>7</td>
<td></td>
<td>9</td>
<td></td>
<td>3</td>
<td></td>
<td>5</td>
<td></td>
<td>15</td>
<td></td>
<td></td>
<td></td>
<td>mod 16</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td>9</td>
<td>6</td>
<td>13</td>
<td>7</td>
<td>3</td>
<td>5</td>
<td>15</td>
<td>2</td>
<td>12</td>
<td>14</td>
<td>10</td>
<td>4</td>
<td>11</td>
<td>8</td>
<td>16</td>
<td></td>
<td></td>
<td>mod 17</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td></td>
<td></td>
<td></td>
<td>11</td>
<td></td>
<td>13</td>
<td></td>
<td></td>
<td></td>
<td>5</td>
<td></td>
<td>7</td>
<td></td>
<td></td>
<td></td>
<td>17</td>
<td></td>
<td>mod 18</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td>10</td>
<td>13</td>
<td>5</td>
<td>4</td>
<td>16</td>
<td>11</td>
<td>12</td>
<td>17</td>
<td>2</td>
<td>7</td>
<td>8</td>
<td>3</td>
<td>15</td>
<td>14</td>
<td>6</td>
<td>9</td>
<td>18</td>
<td>mod 19</td>
</tr>
</table>
=== Exercices sur les nombres premiers entre eux et PGDC ===
1.
:1.
::
::<table border="1" cellpadding="2">
{| border="1" cellpadding="2"
<tr>
|-----
<th>Plus petit</th> <th>Plus grand</th>
! Plus petit
</tr>
! Plus grand
<tr>
|-----
<td>5050</td> <td>5051</td>
| 5050 || 5051
</tr>
|-----
<tr>
| 1 || <td>1</td> <td>5050</td>
|-----
</tr>
| 0 || '''1'''
<tr>
|}
<td>0</td> <td>'''1'''</td>
</tr>
</table>
::5050 et 5051 sont premiers entre eux
:2.
::
::<table border="1" cellpadding="2">
{| border="1" cellpadding="2"
<tr>
|-----
<th>Plus petit</th> <th>Plus grand</th>
! Plus petit
</tr>
! Plus grand
<tr>
|-----
<td>59</td> <td>78</td>
| 59 || 78
</tr>
|-----
<tr>
| 19 || 59
<td>19</td> <td>59</td>
|-----
</tr>
| 2 <tr>|| 19
|-----
<td>2</td> <td>19</td>
| 1 </tr>|| 2
|-----
<tr>
| 0 || '''1'''
<td>1</td> <td>2</td>
|}
</tr>
<tr>
<td>0</td> <td>'''1'''</td>
</tr>
</table>
::59 et 79 sont premiers entre eux
:3.
::
::<table border="1" cellpadding="2">
{| border="1" cellpadding="2"
<tr>
|-----
<th>Plus petit</th> <th>Plus grand</th>
! Plus petit
</tr>
! Plus grand
<tr>
|-----
<td>111</td> <td>369</td>
| 111 || 369
</tr>
|-----
<tr>
| 36 || 111
<td>36</td> <td>111</td>
|-----
</tr>
| 3 <tr>|| 36
|-----
<td>3</td> <td>36</td>
| 0 || '''3'''
</tr>
|}
<tr>
<td>0</td> <td>'''3'''</td>
</tr>
</table>
::111 et 369 ne sont pas premiers entre eux
:4.
::
::<table border="1" cellpadding="2">
{| border="1" cellpadding="2"
<tr>
|-----
<th>Plus petit</th> <th>Plus grand</th>
! Plus petit
</tr>
! Plus grand
<tr>
|-----
<td>2021</td> <td>4032</td>
| 2021 || 4032
</tr>
|-----
<tr>
| 2011 || 2021
<td>2011</td> <td>2021</td>
|-----
</tr>
| 10 || 2011
<tr>
|-----
<td>10</td> <td>2011</td>
| 1 </tr>|| 10
|-----
<tr>
| 0 || '''1'''
<td>1</td> <td>10</td>
|}
</tr>
<tr>
<td>0</td> <td>'''1'''</td>
</tr>
</table>
::2021 et 4032 sont premiers entre eux
2.Nous calculerons d'abord le PGDC pour toutes les combinaisons
 
<table border="1" cellpadding="2">
{| border="1" cellpadding="2"
<tr>
|-----
<th>Plus petit</th> <th>Plus grand</th>
! Plus petit
</tr>
! Plus grand
<tr>
|-----
<td>15</td> <td>510</td>
| 15 || 510
</tr>
|-----
<tr>
| 0 || <td>0</td> <td>'''15'''</td>
|}
</tr>
 
</table>
<table{| border="1" cellpadding="2">
|-----
<tr>
! <th>Plus petit</th> <th>Plus grand</th>
! Plus grand
</tr>
|-----
<tr>
| 15 || 375
<td>15</td> <td>375</td>
|-----
</tr>
| 0 || '''15'''
<tr>
|}
<td>0</td> <td>'''15'''</td>
 
</tr>
{| border="1" cellpadding="2"
</table>
|-----
<table border="1" cellpadding="2">
! Plus petit
<tr>
! <th>Plus petit</th> <th>Plus grand</th>
|-----
</tr>
| 375 || 510
<tr>
|-----
<td>375</td> <td>510</td>
| 135 || 375
</tr>
|-----
<tr>
| 105 || 135
<td>135</td> <td>375</td>
|-----
</tr>
| 30 || 105
<tr>
|-----
<td>105</td> <td>135</td>
| 15 || 30
</tr>
|-----
<tr>
| 0 || '''15'''
<td>30</td> <td>105</td>
|}
</tr>
<tr>
<td>15</td> <td>30</td>
</tr>
<tr>
<td>0</td> <td>'''15'''</td>
</tr>
</table>
:Le PGDC pour toute combinaison des nombres est 15 donc le PGDC est 15 pour les trois nombres.
Récupérée de « https://fr.wikibooks.org/wiki/Approfondissements_de_lycée/ES_Premiers »