how to check that 2814749767109 and 59482661568307 are relatively prime

how to check that 2814749767109 and 59482661568307 are relatively prime

GCD(59482661568307, 2814749767109 ) with Euclidean algorithm:

$$\frac{59482661568307 }{2814749767109} = \frac{59482661568307 -21*2814749767109}{2814749767109} = \frac{372916459018 }{2814749767109}\\\\ = \frac{372916459018 }{2814749767109} = \frac{372916459018 }{2814749767109-7*372916459018} = \frac{372916459018 }{204334553983} \\\\ = \frac{372916459018 }{204334553983} = \frac{372916459018-204334553983 }{204334553983} = \frac{168581905035 }{204334553983}\\\\ = \frac{168581905035 }{204334553983} = \frac{168581905035 }{204334553983-168581905035} = \frac{168581905035 }{35752648948} \\\\ \dots$$

a                             b                                q           r

59482661568307  2814749767109     21          372916459018

2814749767109      372916459018       7           204334553983

372916459018        204334553983       1           168581905035

204334553983        168581905035       1             35752648948

168581905035          35752648948       4              25571309243

35752648948             25571309243       1             10181339705

25571309243             10181339705       2               5208629833

10181339705               5208629833       1               4972709872

5208629833                 4972709872       1                 235919961

4972709872                   235919961     21                   18390691

235919961                       18390691     12                   15231669

18390691                         15231669       1                     3159022

15231669                           3159022       4                     2595581

3159022                             2595581       1                       563441

2595581                               563441       4                       341817

563441                                 341817       1                       221624

341817                                 221624       1                       120193

221624                                 120193       1                       101431

120193                                 101431       1                         18762

101431                                   18762       5                           7621

18762                                       7621       2                           3520

7621                                         3520       2                              581

3520                                           581       6                                34

581                                               34      17                                 3

34                                                    3     11                                   $$\textcolor[rgb]{1,0,0}{1}$$

3                                                      1       3                                  0

GCD = 1, so 2814749767109 and 59482661568307 are relatively prime

Find the prime factors of each:

$${factor}{\left({\mathtt{2\,814\,749\,767\,109}}\right)} = {\mathtt{7}}{\mathtt{\,\times\,}}{\mathtt{65\,111}}{\mathtt{\,\times\,}}{\mathtt{6\,175\,717}}$$

$${factor}{\left({\mathtt{59\,482\,661\,568\,307}}\right)} = {\mathtt{7\,577\,329}}{\mathtt{\,\times\,}}{\mathtt{7\,850\,083}}$$

No factor in common, therefore they are relatively prime.

.