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The original number is divisible by 3 (or 9) if and only if the sum of its digits is divisible by 3 (or 9). Adding the digits of a number up, and then repeating the process with the result until only one digit remains, will give the remainder of the original number if it were divided by nine (unless that single digit is nine itself, in which ...
If the digit 9 is ignored when summing the digits, the effect is to "cast out" one more 9 to give the result 12. More generally, when casting out nines by summing digits, any set of digits which add up to 9, or a multiple of 9, can be ignored. In the number 3264, for example, the digits 3 and 6 sum to 9.
d() is the number of positive divisors of n, including 1 and n itself; σ() is the sum of the positive divisors of n, including 1 and n itselfs() is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n
Demonstration, with Cuisenaire rods, of the first four highly composite numbers: 1, 2, 4, 6. A highly composite number is a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N.
The number 18 is a harshad number in base 10, because the sum of the digits 1 and 8 is 9, and 18 is divisible by 9. The Hardy–Ramanujan number (1729) is a harshad number in base 10, since it is divisible by 19, the sum of its digits (1729 = 19 × 91). The number 19 is not a harshad number in base 10, because the sum of the digits 1 and 9 is ...
9 is the fourth composite number, and the first odd composite number. 9 is also a refactorable number. [2] Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers in decimal, a method known as long ago as the 12th century. [3]
Digit sums and digital roots can be used for quick divisibility tests: a natural number is divisible by 3 or 9 if and only if its digit sum (or digital root) is divisible by 3 or 9, respectively. For divisibility by 9, this test is called the rule of nines and is the basis of the casting out nines technique for checking calculations.
360 is divisible by the number of its divisors , and it is the smallest number divisible by every natural number from 1 to 10, except 7. Furthermore, one of the divisors of 360 is 72, which is the number of primes below it. 360 is the sum of twin primes (179 + 181) and the sum of four consecutive powers of three (9 + 27 + 81 + 243).