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The last two digits form a number that is divisible by 4. [2] [3] 40,832: 32 is divisible by 4. If the tens digit is even, the ones digit must be 0, 4, or 8. If the tens digit is odd, the ones digit must be 2 or 6. 40,832: 3 is odd, and the last digit is 2. The sum of the ones digit and double the tens digit is divisible by 4.
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
32 is the fifth power of two (), making it the first non-unitary fifth-power of the form where is prime. 32 is the totient summatory function over the first 10 integers, [1] and the smallest number with exactly 7 solutions for ().
1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd. 1, −1, and are known as the trivial divisors of .
Formally, a regular number is an integer of the form , for nonnegative integers , , and .Such a number is a divisor of (⌈ / ⌉,,).The regular numbers are also called 5-smooth, indicating that their greatest prime factor is at most 5. [2]
It is the smallest number divisible by every natural number from 1 to 10, except 9. It is the largest number k such that all coprime quadratic residues modulo k are squares. In this case, they are 1, 121, 169, 289, 361 and 529.
Demonstration, with Cuisenaire rods, that 1, 2, 8, 9, and 12 are refactorable. A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that ().
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).