Which layer of earth's atmosphere contains no water vapor, has an atmospheric pressure less than 0.1 atmosphere, and has an air temperature that increases with altitude?

Thermosphere Let's look at the layers from the bottom to the top and see where the first match is. Troposphere - Lowest layer. Temperature usually declines with increasing altitude. Almost all the water vapor is contained in this layer... This is not the right layer. Lots of water vapor and decreasing temperatures. So let's continue. Stratosphere - Second-lowest layer of Earth's atmosphere. Temperatures rise with increasing altitude. Pressure ranges from about 200 millibars down to 1 millibar. The stratosphere is almost completely free of clouds. Got a couple of problems here. The pressure is a bit higher than we want, and there are sometimes clouds here. Which means water vapor. Not the right layer, so up we go. Mesosphere - Third layer. Temperatures drop with increasing altitude. Dropping temperatures. Don't want this layer either. Still going up. Thermosphere - Fourth layer. Temperatures increase with increasing altitude. This layer is completely cloudless and free of water vapor. Looks good. No clouds ever and increasing temperature. The pressure is well below 0.1 atmospheres. Looks like this is the answer.

81 is a composite number