924 is a positive integer with several notable arithmetic and combinatorial properties. It is composite, even, and appears in combinatorics as a central binomial coefficient.
Arithmetic facts
- Prime factorization: 924 = 22 · 3 · 7 · 11.
- Number of positive divisors: 24. (If n = p1a1 · p2a2 · ..., then the count is (a1+1)(a2+1)...; here (2+1)(1+1)(1+1)(1+1) = 24.)
- List of positive divisors: 1, 2, 3, 4, 6, 7, 11, 12, 14, 21, 22, 28, 33, 42, 44, 66, 77, 84, 132, 154, 231, 308, 462, 924.
- Sum of all divisors σ(924) = 2,688, so the sum of proper divisors is 2,688 − 924 = 1,764. Because the sum of proper divisors exceeds 924, the number is classified as abundant. The abundance (excess) equals 1,764 − 924 = 840.
- Euler's totient: φ(924) = 240 (computed multiplicatively from the factorization: φ(22)·φ(3)·φ(7)·φ(11) = 2·2·6·10 = 240).
Combinatorial interpretation
924 is a binomial coefficient: it equals C(12, 6) = 12! / (6!·6!). In other words, there are 924 distinct 6-element subsets of a 12-element set. Because it has the form C(2n, n) for n = 6, it is a central binomial coefficient.
Numeral system representations
- Binary (base 2): 11100111002.
- Octal (base 8): 16348.
- Hexadecimal (base 16): 39C16 (0x39C).
Additional remarks
- 924 is divisible by several small primes (2, 3, 7, 11); for example, 924 = 11 · 84.
- Its aliquot sequence begins by mapping 924 → 1764, since the sum of proper divisors of 924 equals 1764. (1764 = 22 · 32 · 72.)
- 924 is not a Harshad (Niven) number in base 10, because it is not divisible by the sum of its decimal digits (9 + 2 + 4 = 15, and 924 is not divisible by 15).