I saw the following sentence:
She's a very youthful 65.
It seems "65" is a countable noun, but can we say "She's a 65"?
I'd appreciate your help.
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Sign up to join this communityNo - it's not idiomatic to say She's a 65, and I don't think it's particularly meaningful to say 65 is a "countable noun" in the cited context.
But it is useful to think of the usage as implying there are many different ways of being 65 - one/some of which look younger than you might expect from the number of years. Effectively, we're being invited to think of "youthful appearance" as being an attribute of some examples of [people at the age of] 65. You could compare this usage to...
It was a chastened Emmanuel Macron who addressed the French nation on December 10th
The Economist, Dec 11th 2018
...where the implication is there are different "manifestations" of Macron - and one he showed in that national address was a chastened / apologetic "version" of himself. And for another version based entirely on numbers, I found...
Many a woman is a perfect 36-24-36
The Comic Encyclopedia (1978)*,
(Where those numbers represent the idealised "hourglass" figure for a woman, expressed as bust-waist-hip measurements in inches.)
One key point about this usage is that there must be some kind of adjectival element before the "head noun" (65, Emmanuel Macron, 36-24-36), to distinguish the version being referred to from other instances of those "nouns" that don't have the adjectivally-specified quality.
Unless it makes sense to interpret the head noun as a short form of something else. Which works with a 36-24-36 = a 36-24-36 figure, but not really with a 65 = a 65-year-old - and there's simply no way to do this with Emmanuel Macron.
As regards "parts of speech", I don't think that's a very useful concept in contexts like this. But given the article (a youthful 65), if you feel you must assign some kind of label, it could only be classified as a noun phrase.
You might find the ELU question Why is it “a defeated Napoleon”, not “the defeated Napoleon” who rode off the battlefield and into exile? useful. Note that the usage is fairly unlikely in colloquial speech - it's a literary/oratorial device, that you're actually more likely to find in newspapers and magazine articles today.
She's a 65 would mean:
she is literally the number 65.
she is rated a 65 in some sort of contest or evaluation.
It may help if you think of She's 65 as short for She's 65 years old.
There's 65 ... and there's 65. There's a decrepit 65 and a spry 65.
"countable" is such a crappy way of labeling the phenomenon. a can be an individuation or sampling marker.
Each adjective creates a subset of 65. a culls a member of the subset.