which is magical number from 0 to 9?
Answers
Step-by-step explanation:
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. It is the first composite lucky number, along with the first composite odd number and only single-digit composite odd number.
9 is the only positive perfect power that is one more than another positive perfect power, by Mihăilescu's Theorem.
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.
Since 9 = 321, 9 is an exponential factorial.
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:
2 × 9 = 18 (1 + 8 = 9)
3 × 9 = 27 (2 + 7 = 9)
9 × 9 = 81 (8 + 1 = 9)
121 × 9 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9)
234 × 9 = 2106 (2 + 1 + 0 + 6 = 9)
578329 × 9 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27; 2 + 7 = 9)
482729235601 × 9 = 4344563120409 (4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45; 4 + 5 = 9)
There are other interesting patterns involving multiples of nine:
12345679 × 9 = 111111111
12345679 × 18 = 222222222
12345679 × 81 = 999999999
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.
The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples:
The sum of the digits of 41 is 5, and 41 − 5 = 36. 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 35967930 is 3 + 5 + 9 + 6 + 7 + 9 + 3 + 0 = 42, and 35967930 − 42 = 35967888. The digital root of 35967888 is 3 + 5 + 9 + 6 + 7 + 8 + 8 + 8 = 54, 5 + 4 = 9.
Casting 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.
Six recurring nines appear in the decimal places 762 through 767 of π, see Six nines in pi.
If dividing a number by the amount of 9s corresponding to its number of digits, the number is turned into a repeating decimal.
Example
274
/
999
= 0.274274274274...)
There are nine Heegner numbers.[6]