Réponse :
Bonsoir,
Explications étape par étape
1)
[tex]d_1,d_2,d_3\ \in\ \mathbb{N}^*:\\d_1\ | \ d_2\ \exists \ k_1\ \in\ \mathbb{N}^*:\ d_2=k_1*d_1\\d_2\ | \ d_3\ \exists \ k_2\ \in\ \mathbb{N}^*:\ d_3=k_2*d_2\\\Longrightarrow\ \exists \ k_2*k_1\ \in\ \mathbb{N}^*:\ d_3=(k_2*k_1)*d_1\\\Longrightarrow\ d_1\ |\ d_3\\[/tex]
2)
[tex]a) \\\\a,b\ \in\ \mathbb{N}^*, \ b > a, \ d | a, \ d | b\\\\d\ | \ a\ \exists \ k_1\ \in\ \mathbb{N}^*:\ a=k_1*d\\d\ | \ b\ \exists \ k_2\ \in\ \mathbb{N}^*:\ b=k_2*d\\\Longrightarrow\ \exists \ k_2-k_1\ \in\ \mathbb{N}^*:\ b-a=(k_2-k_1)*d\\\Longrightarrow\ d\ |\ b-a\\[/tex]
b)
[tex]d\ | \ a\\d\ | \ b\\\\b=a*q+r\\r=b-a*q \\\\d | b\ et\ d| a*q \Longrightarrow\ r | b-a*q=r\\\\\begin {array} {c|c|c|c|c|c|c|c}q&&20&2&1&1&1&78\\--&--&--&--&--&--&--&--\\D&25632&1258&472&314&158&156&2\\r&472&314&158&156&2&0&\\\end{array}\\\\\\\boxed{pgcd(25632,1258)=78}[/tex]
