Nine is an integer multiple of three.
Nine is an integral multiple of three.
Which is more common?
If both are accepted, what's the subtle difference between them?
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As a rule if we have a single-word noun, and if there is a single-word adjective with the same meaning that we get from using the noun as the modifier (especially a directly related one), then while using the noun as a modifier for another noun is acceptable, using the adjective is the more popular decision, and tends to read as more graceful English to many.
In this case however:
So while there's no difference in meaning, and you certainly can use integral, there is a good argument in favour of integer here.
I favour integral, but then that's precisely because I'm the sort of person who spends more time on ELU than the mathematics stack-exchange. I'm not the target audience.