72 Hour Visa. Do we qualify for the 72 hours without a visa?

I am a US citizen. I have a flight from the US to Beijing which lands at 15:15 on July 21st. Our flight from Beijing to Ulaanbaatar, Mongolia is on July 23rd at 9:30.

We fly back to Beijing from Mongolia and arrive on August 2nd at 1:00. Our departing flight from Beijing to the US is at 3:35 on August 2nd.