Six recurring nines appear in the decimal places 762 through 767 of π, see Six nines in pi. The digital root of 35967888 is 3 + 5 + 9 + 6 + 7 + 8 + 8 + 8 = 54, 5 + 4 = 9.Ĭasting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers known as long ago as the 12th century. The digital root of 36 is 3 + 6 = 9, which, as explained above, demonstrates that it is divisible by nine. The sum of the digits of 41 is 5, and 41 − 5 = 36.The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Another consequence of 9 being 10 − 1, is that it is also a Kaprekar number. In base- N, the divisors of N − 1 have this property. 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. There are other interesting patterns involving multiples of nine: 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: In base 10, a positive number is divisible by 9 if and only if its digital root is 9.
A group of nine of anything is called an ennead. Ī polygon with nine sides is called a nonagon or enneagon. Since 9 = 3 2 1, 9 is an exponential factorial. All subsequent squares of this form are odd. It is the second non-unitary square prime of the form ( p 2) and the first that is odd. 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.ĩ is the highest single-digit number in the decimal system. It is the first composite lucky number, along with the first composite odd number and only single-digit composite odd number.ģ times 3 is one more than 2 times 2 times 2.
It is 3 times 3 and hence the third square number. 9 is a composite number, its proper divisors being 1 and 3.
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