Have to is NOT an auxiliary verb. We need do/does/did to make negatives and questions in the present and past.
We can use have to in all the verb forms (present, past and perfect tenses, gerund, to infinitive, etc.)
It is used to describe an obligation, a rule, or something that is necessary.
In spoken informal English, you can often use ‘ve got to instead of have to.
Must is a modal verb, and it has a present form, which can be used to talk about the present or future. The negative form is must not or mustn’t, and the question is must I, must you, etc.
It is used to describe something that the speaker thinks is necessary to do.
We can also use must to express strong advice.
Must only has a present form, so we need to use have to for all other verb forms (past, future, perfect forms, infinitive, etc.).
We normally use have to for questions. Questions with must are not common.
Regarding the meaning, sometimes there is little difference between have to and must.
But we normally use have to when there’s an external obligation and must when the obligation comes from the opinion of the speaker.
Don’t have to and mustn’t have opposite meanings. We use don’t have to when we don’t need to do something, when there’s no obligation, and we use mustn’t to talk about prohibition, when there is obligation not to do something.
If you don’t have to do something, it means that you can do it if you want, but you don’t need to do it.
We can use can’t or be not allowed to instead of mustn’t.
Should is used to give advice or an opinion about what we think is right or wrong.
Should is not as strong as must or have to.
Ought to has the same meaning as should, although it is more formal and not as common.
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