Grammatical number of numeral 4

Question

"Существует 4 фактора, от которых зависит наше здоровье".

Here the grammatical number is quite confusing.

Nominative 4 (четыре) is followed by genitive singular фактора. No doubt here. If "4" takes another case (ex.: genitive четырёх) than "фактор" takes the corresponding plural case (becoming четырёх факторов then). So far so good.

But, what's the grammatical number of четыре itself, if it's a noun rather than just a numeral?

I mean, "существуЕТ (third person singular) четыре фактора". Is "четыре" a singular noun?

And then, "от которЫХ" (genitive plural) when "фактора" is genitive singular. Does the relative pronoun refer directly to the fact that we are talking about multiple factors, regardless of the genitive singular which is, say, a grammatical "whim" of the Russian language?

Answer 1

The sentence you quoted is grammatically correct.

In Russian, cardinal numerals do not have a grammatical number and whenever such numeral appears as the subject of a sentence, the predicate can assume either number. The choice is not completely arbitrary though:

• singular is preferred, when the emphasis is on the number itself or the subject(s) were passive or they all participated in the action in the same way:
  • В русском имеется 6 падежей. (emphasis on the number)
    
    In the past tense, singular neuter is used (-о):
  • В результате урагана упало двадцать деревьев. (trees were passive)
  • В авиакатастрофе погибло 200 человек. (single event, people were passive victims)
  • На лекцию пришло пятьдесят человек, а не пятьдесят два. (emphasis on the number)
• plural is preferred, when you want to emphasise the subjects' active involvement in the action or the action was multiple events or was performed by each subject in their own way:
  • Сто аспирантов пишут диссертации (each is writing their own)
  • Двадцать один участник обсудили проблему (Yes, the predicate is plural despite the seemingly singular subject "двадцать один". Each participant was actively involved in their own way.)

'Existence verbs' (существует, имеется, наличиствует, есть) are a common example of a passive action (you don't need to do anything to exist) and thus singular is preferred in

• Существует 4 фактора, от которых зависит наше здоровье.

You can say "Существуют 4 фактора..." if you want to emphasise the uniqueness of each factor.

The above is based on the article Единственное и множественное число сказуемого (thanks to @БаянКупи-ка for the link)

As for которых, the rules are different for numeral-adjective agreement:

• Существует 1 основной фактор, от которого зависит наше здоровье.
• Существует 3 основных фактора, от которых зависит наше здоровье.
• Существует 5 основных факторов, от которых зависит наше здоровье.
• Существует 21 основной фактор, от которых зависит наше здоровье.

Answer 2

4 фактора is the correct agreement between a numeral and a noun. To have the noun assume the form of Genitive plural факторОВ, 4 would have to be construed as четвёрка, that is as another noun.

But we have divergence between существуЕТ and которЫХ.

Thus in my opinion this is a case of erroneous agreement of a predicate with a subject expressed with a numeral.

The verb has to assume plural form

СуществуЮТ 4 фактора, от которЫХ ...

In vernacular and in grammarless speech the rules of agreement with numerals in particular are very often overlooked and verbs assume neutral inflection, not least because these rules are very vague, like those described in the section Согласование сказуемого с числительными of an article on gramota.ru