Concept
24 (twenty-four) is the natural number following 23 and preceding 25. 24 is an even composite number, with 2 and 3 as its distinct prime factors. It is the first number of the form 2^qq, where q is an odd prime. It is the smallest number with at least eight positive divisors: 1, 2, 3, 4, 6, 8, 12, and 24; thus, it is a highly composite number, having more divisors than any smaller number. Furthermore, it is an abundant number, since the sum of its proper divisors (36) is greater than itself, as well as a superabundant number. 24 is the smallest 5-hemiperfect number, as it has a half-integer abundancy index: 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = 5/2 × 24 24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24. 24 is a practical number, since all smaller positive integers than 24 can be represented as sums of distinct divisors of 24. 24 is a Harshad number, since it is divisible by the sum of its digits in decimal. 24 is a refactorable number, as it has exactly eight positive divisors, and 8 is one of them. 24 is a twin-prime sum, specifically the sum of the third pair of twin primes . 24 is a highly totient number, as there are 10 solutions to the equation φ(x) = 24, which is more than any integer below 24. 144 (the square of 12) and 576 (the square of 24) are also highly totient. 24 is a polite number, an amenable number, an idoneal number, and a tribonacci number. 24 forms a Ruth-Aaron pair with 25, since the sums of distinct prime factors of each are equal (5). 24 is a compositorial, as it is the product of composite numbers up to 6. 24 is a pernicious number, since its Hamming weight in its binary representation (11000) is prime (2). 24 is the third nonagonal number. 24's digits in decimal can be manipulated to form two of its factors, as 2 * 4 is 8 and 2 + 4 is 6. In turn 6 * 8 is 48, which is twice 24, and 4 + 8 is 12, which is half 24. 24 is a congruent number, as 24 is the area of a right triangle with a rational number of sides.