The phrase "mechanically fastened another button of his shirt and did not reply" shows that "absently" means not paying attention.
<h3>The meaning of "absently"</h3>
It is common for authors to use context clues when they introduce a word readers may not know. Context clues are pieces of information that help readers figure out the meaning of that word.
In the passage we are analyzing here, it is possible to infer the meaning of "absently" when we read the phrase that says "mechanically fastened another button." When we do something mechanically, we do it automatically, without giving it much thought.
In other words, we do it without paying attention. That is precisely what "absently" mean. Thus, the phrase helps us figure out the meaning of "absently."
Learn more about context clues here:
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The last one
names of a student’s favorite celebrities
it is unrelating to school and will not interest many people
B because in the context it would have been seen more dramatic