981 is a three-digit natural number that follows 980 and precedes 982. It is an odd composite integer notable for a small number of simple arithmetic properties: its prime factorization is 3^2 · 109, it is divisible by 3 and 9, and it can be expressed as the sum of two squares (30^2 + 9^2 = 981). In everyday contexts the symbol "981" can denote the integer itself or be used as a label—such as a year number, an item identifier, or part of a serial code.

Mathematical properties

Key numerical attributes of 981 include:

  • Prime factorization: 981 = 3^2 × 109.
  • Divisors: 1, 3, 9, 109, 327, 981 (six positive divisors).
  • Sum of divisors: σ(981) = 1 + 3 + 9 + 109 + 327 + 981 = 1430; the sum of proper divisors is 449, which is less than 981, so 981 is a deficient number.
  • Euler totient: φ(981) = φ(3^2)·φ(109) = 6 × 108 = 648.
  • Representation as sum of squares: 981 = 30^2 + 9^2, a consequence of its prime factors having suitable exponents and congruence classes.

Numeral-system representations and notation

How 981 appears in common number systems and notations:

  • Base 10: 981
  • Binary (base 2): 1111010101
  • Hexadecimal (base 16): 0x3D5
  • Roman numerals: CMLXXXI (900 = CM, 80 = LXXX, 1 = I)

Its digital sum in base 10 is 9 + 8 + 1 = 18, which explains divisibility by 9 and by 3. Because 3 appears with exponent 2 in the factorization, 981 is not squarefree.

Context, uses and cultural notes

As with any integer, "981" is used as an ordinal or identifier in many practical contexts: years (e.g., 981 AD or 981 BC when referring to calendar years), model numbers, page or entry numbers, and catalogue references. In recreational mathematics it can appear in problems about representations, divisibility, totients and sums of squares. The Roman-numeral form CMLXXXI appears in inscriptions or stylistic references where Roman numerals are used.

Although not especially prominent among integers, 981 provides simple examples for teaching multiplicative functions (like σ and φ), modular divisibility rules (by 3 and 9), and the theorem on expressing integers as sums of two squares when primes of form 4k+3 occur with even exponents.