A prime number is a number greater than one that has no positive divisors other than 1 and itself.
Although you can't divide 1 by anthing but 1, the number 1 doesn't meet the second qualifiation "a number that is greater than 1"
That's why 1 isn't a prime number.
A little off topic of the actual question, here's a chart of all the prime numbers 1-100. Highlighted means it's a prime number. :)

Definition paraphrased from Wikipedia: http://en.wikipedia.org/wiki/Prime_number
Picture source: https://missmanganssciencesite.pbworks.com/w/page/32589311/Prime%20Number%20Determination%20Method