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At about the same time Franklin wrote a last letter to his friend asking him...

I read this in a book. If I change it using the like:
At about the same time Franklin wrote the last letter to his friend asking him…

What difference will it make? Actually I'm wondering the original sentence should use the instead of a as last means definite.

Thanks in advance.
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The original is natural. It suggests either that he wrote last letters to several friends (and this is one of them, but we don't know whether it was 'the' last one he wrote to anyone) or that this is a sort of reclassification, where 'last' becomes a kind of letter.
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Thanks as always Mister Micawber!