Overview

272 is a natural number that follows 271 and precedes 273. It is even and composite, with a simple prime factorization that reveals much of its basic arithmetic structure. Because of that factorization, several standard number‑theoretic functions take neat values for 272.

Arithmetic properties

Prime factorization: 272 = 2^4 × 17. From this you can derive a number of elementary invariants:

  • Divisors (10 total): 1, 2, 4, 8, 16, 17, 34, 68, 136, 272.
  • Sum of all divisors σ(272) = 558; sum of proper divisors = 286.
  • Euler's totient φ(272) = 128 (indeed 272×(1−1/2)×(1−1/17) = 128), a power of two.
  • Number of positive divisors τ(272) = (4+1)(1+1) = 10.

Because the sum of proper divisors (286) exceeds 272, the number is classified as abundant. It is not a weird number: 272 can be written as a sum of some of its proper divisors, for example 272 = 136 + 68 + 34 + 16 + 17 + 1.

272 begins an aliquot sequence 272 → 286 → 218 → 112 → ... where each term is the sum of proper divisors of the previous term. The appearance of φ(272)=128 is notable because it is a pure power of two, reflecting the high 2‑power in 272's factorization. With ten divisors, 272 is not refactorable (10 does not divide 272).

Representations and simple examples

Common representations of 272 include binary 100010000 (256+16), hexadecimal 0x110, and the Roman numeral CCLXXII. Its divisibility by 16 and 17 makes it a convenient example when illustrating modular arithmetic or tests for divisibility by 17. Because 272 = 16×17, it is the 17th multiple of 16 and the 16th multiple of 17.

History and other contexts

The symbol "272" can also denote years. In the late Roman period, events recorded in 272 AD are associated with the reign of Emperor Aurelian and the Roman reconquest of eastern territories; the label is mainly a chronological reference rather than a mathematical property. Outside mathematics, the number may appear in identifiers, model numbers, or counts without intrinsic numerical significance.

In summary, 272 is a modestly interesting composite integer: small, abundant yet semiperfect, with a compact factorization that produces tidy arithmetic function values and straightforward representations in multiple bases.