I'd be careful about "keeping everybody on the same page". If you know they'd appreciate it, then sure. But a significant portion of the population do not like being fact checked. They assume malicious intent (not realizing you factcheck everyone and not just them).
Below is an email I once got. The context: A bunch of us were having a social conversation after an event. A professor made a claim about how hot it would get in his country. It struck me and another one as off because the number was a bit higher than the world record. We hinted at it but he insisted we were wrong.
Some hours later, at home, my friend fact checked and sent a polite email pointing out that the highest ever recorded temperature in his country was a few degrees lower than his claim. His response:
"If I were you i would not have spent a minute doing that unless you want to prove a point: I was a liar. ... At last, that's why I do not hang around with you guys."
People will jump to conclusions about your motives.