Overview

Seventy-two is an even whole number that follows 71 and precedes 73. In English its name combines "seventy" and "two." As an integer it occurs frequently in arithmetic, geometry, measurement and cultural contexts.

Mathematical properties

In prime factorization 72 = 2^3 × 3^2. It is a composite, even number with 12 positive divisors. Because it is the product of two consecutive integers (8 × 9), 72 is a pronic or oblong number. It is also abundant (the sum of its proper divisors exceeds the number itself) and refactorable (divisible by its count of divisors).

  • Positive divisors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
  • Number of divisors: 12. Sum of all divisors: 195; sum of proper divisors: 123.
  • Euler's totient: φ(72) = 24. In common numeral systems: binary 1001000, octal 110, hexadecimal 48, Roman numeral LXXII.

Uses and occurrences

Numerous practical and geometric occurrences make 72 a useful benchmark. A regular pentagon has central angles of 72° (360°/5), a value important in design and tiling. In typography and digital graphics, PostScript defines 72 points to an inch, a historic basis for point-to-inch conversions. In sports, many professional golf tournaments use 72 holes (four rounds of 18) as the standard event length. The chemical element with atomic number 72 is hafnium.

Notable distinctions and cultural notes

Beyond arithmetic, 72 appears in a variety of cultural and traditional lists and measures across different societies. Mathematically it is a semiperfect number because some subsets of its divisors sum to 72 (for example 36+24+12). Readers can find a basic reference to 72 as an even number or explore deeper number-theory resources for proofs and sequences.