Sixty-six is the natural integer that follows sixty-five and precedes sixty-seven. It is an integer often written as "66" in decimal form. As an even number, it is divisible by 2 and several other positive integers; one of its elementary divisibility facts is that it is divisible by 1 (as are all whole numbers).
Basic arithmetic and divisors
In prime-factor terms, 66 = 2 × 3 × 11. Because it is the product of three distinct primes it belongs to special classes of integers: it is composite, sphenic, and even. The complete set of positive divisors of 66 is listed below.
- Divisors: 1, 2, 3, 6, 11, 22, 33, 66
- Prime factorization: 2 × 3 × 11
- Number type: even, composite, sphenic, abundant (sum of proper divisors = 78 > 66)
Figurate identities and combinatorics
Sixty-six is the 11th triangular number because 1+2+...+11 = 66. It is also the 6th hexagonal number, showing it appears among more than one family of figurate numbers. In combinatorics 66 arises as the binomial coefficient C(12,2), which counts unordered pairs chosen from 12 items — for example, 12 people producing 66 distinct handshakes if every pair shakes once.
Representations and notable occurrences
Common numeral forms include Roman numerals (LXVI) and binary (1000010). In chemistry, the atomic number 66 identifies the element dysprosium. The two-digit repetition makes 66 visually distinctive and it appears frequently in cultural or infrastructural names (for instance, road numbers and product model names).
Although a simple integer, 66 connects several elementary topics in number theory (divisibility, factorization, figurate numbers) and serves as a convenient example for combinatorial counting and classroom problems. Its combination of small prime factors and triangular/hexagonal identities makes it a compact illustration of how arithmetic and geometry can intersect.