9
9 is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. Nine is a Motzkin number.[1] It is the first composite lucky number, along with the first composite odd number and only single-digit composite odd number.
3 times 3 is one more than 2 times 2 times 2. Thus, 9 is a positive perfect power that is one more than another positive perfect power, and it can be proved by Mihăilescu's Theorem that 9 is the only number having this property.
9 is the highest single-digit number in the decimal system. It is the second non-unitary square prime of the form (p2) and the first that is odd. All subsequent squares of this form are odd.
A polygon with nine sides is called a nonagon or enneagon.[3] A group of nine of anything is called an ennead.
In base 10, a positive number is divisible by 9 if and only if its digital root is 9.[4] That is, if any natural number is multiplied by 9, and the digits of the answer are repeatedly added until it is just one digit, the sum will be nine:
This works for all the multiples of 9. n = 3 is the only other n > 1 such that a number is divisible by n if and only if its digital root is divisible by n. In base-N, the divisors of N − 1 have this property. Another consequence of 9 being 10 − 1, is that it is also a Kaprekar number.
