Overview

54 is a positive natural number that follows 53 and precedes 55. It is an even number and a composite integer commonly encountered in arithmetic, counting, and everyday contexts.

Mathematical characteristics

Its prime factorization is 2 × 3^3. The full set of positive divisors is 1, 2, 3, 6, 9, 18, 27 and 54. The sum of all divisors (the sigma function) is 120, which equals 5!, and the sum of proper divisors (1 + 2 + 3 + 6 + 9 + 18 + 27) is 66, larger than 54; therefore 54 is classified as an abundant number. Because a subset of its proper divisors (for example 27 + 18 + 9) sums to 54, it is also semiperfect.

Representations and divisibility

In Roman numerals 54 is written LIV. Common positional representations include binary 110110, octal 66 and hexadecimal 36. It is divisible by 2, 3, 6, 9 and 27; divisibility by 3 and 9 can be checked by the digit-sum rule (5 + 4 = 9). 54 is a Harshad (Niven) number in base 10 because it divides evenly by the sum of its digits.

The aliquot sum of 54 is 66, which begins an aliquot sequence 54 → 66 → 78 → … of composite values. It is not a prime, not a perfect number, and not a triangular or square integer. The arithmetic properties tied to its factorization make 54 a useful example in elementary number theory and teaching divisibility concepts.

Cultural and scientific notes

Outside mathematics, 54 appears as the atomic number of the noble gas xenon and is the number of cards in a standard playing-card pack when two jokers are included. It also appears in cultural references and names, reflecting its recognizability in everyday life.

  • Prime factors: 2, 3 (with multiplicity: 2 × 3^3)
  • Divisors: 1, 2, 3, 6, 9, 18, 27, 54
  • Sigma(54) = 120 = 5!