There are so many different ways to factor 36. That's part of its appeal. But there's only one way to factor 37.
But 37 is also a prime number, and that makes it special in its own right.
According to the Wikipedia, 37 is the first irregular prime, the third unique prime, and the fifth lucky prime.
37 is thus far the only number I know of that has a web site dedicated to it. Tom Magliery put up thirty-seven.org to list down all things related to 37.
According to Tom:
It does seem that 37 occurs a lot. You hear it a lot in conversation: people choose it as an "arbitrary" number of something. It turns up in writing, such as in columns by Dave Barry and Joe Bob Briggs. It appears in comic strips quite a bit. It has appeared in many movies, both accidentally and on purpose. It turns up on lots of personalized license plates.
It may be that 37 gets used a lot because it somehow "feels random". I imagine psychologists pondering theories and performing experiments on destitute grad students, to figure out why people might choose this number over a less random-sounding number like 36.
I'd list down some of the factoids but it's better if you visit his site. Oh, but do take some things with a grain of salt.
Pictured above is Open Cluster M37, also known as Messier Object 37. It is the richest open cluster in the Auriga constellation. It was discovered by Giovanni Batista Hodierna before 1654. With an average telescope about 150 stars can be identified in it.
And 37 is how old I am today.