Bonjour

1- He doesn't have to > He needn't to buy another one.

do not need = pas nécessaire (puisqu'il en a déjà deux)

2- is not permitted > You may not chew gum in the classroom.

 may not = n'est pas permis; interdiction catégorique, règlement

3- is allowed = You may take  one of each.

may = possibilité

4- You've got to > You must remember their name.

   have got to = must = obligation

5- I'm sure > He must be in here.

(i'm sure )= he must = forte probabilité

6- Why don't you try..> You should try cycling.

should = conseil

7- Why not go with them ?  >> Shall we go with them?

shall = suggestion

8- It's forbidden > You mustn't talk to the prisoners.

must not = interdiction totale (forbidden)

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bonne journée☺☺☺