Any state that won wars because of their numbers alone? Russia.

And I'd say it was the Roman tactics and not numbers (Rome started as a city-state after all) that first got them out of the "hellhole" that was the constant and mostly fruitless fighting between various Hellenistic states.

Ironically, Rome had tendency to lose battles when they amassed large armies (like at Cannae and Carrhae).

I wouldn't say that the Romans won wars because of their numbers. It merely allowed them a buffer. Romans were known for their willpower. They were usually repulsed upon first encounter with enemies but they quickly coagulated and returned with a better understanding. When you look at the things they have to endure, Teutoberg forrest against the germans, Carrhae against the Parthians, and then the annihilation of legion after legion against Spartacus on Italian soil....... But at the end of the day Rome would hold, they took military fiascos that usually ended nations again and again. Its like it was ingrained in them, no wonder they are known for their marvelous architecture and infrastructure.
and yeah, Id have to say Russia However, post Roman Empire/early middle ages was very numbers oriented. It was basically my kingdom versus your kingdom, we smash together until sundown and the next day we go again until one of us gives up

I had to remove this question because the way it was phrased opened it up to answers from multiple time periods. We have a rule here against open-ended or "throughout history" - type questions. These kinds of questions tend to invite lots of trivia and speculation. However, the first part of your question is completely valid. If you want to resubmit it and just limit the question to Rome, that would be fine.