Overview

Ninety-nine is the integer that follows ninety-eight and precedes one hundred. It is the largest two-digit number in base ten, an odd composite, and a decimal repdigit written as "99". For a brief definition see reference.

Basic characteristics and divisors

As an odd composite number, 99 has a small set of positive divisors. It is divisible by 1 and by the primes and composite factors listed below. Its prime factorization is 3^2 × 11, so it is divisible by 3, 9, and 11; therefore it is divisible by 33 as well. Because the sum of its proper divisors (1 + 3 + 9 + 11 + 33 = 57) is less than 99, it is classified as a deficient number.

  • Positive divisors: 1, 3, 9, 11, 33, 99.
  • Notable divisibilities: by 3, by 9, and by 11.

Algebraic and numeric properties

Algebraically 99 = 10^2 − 1 = (10 − 1)(10 + 1) = 9 × 11, which explains its factorization and why it is the largest two-digit repdigit in base ten. The decimal fraction 1/99 produces the repeating two-digit pattern 0.010101..., a useful example when discussing repeating decimals and cyclic patterns. As an odd integer it shares parity properties with other odd numbers and appears in many elementary identities.

Representations and notations

Common numeral representations of 99 include binary 1100011, octal 143, hexadecimal 63, and Roman numerals XCIX. Its digits sum to 18, which is divisible by 9; this provides a quick divisibility test for 9 and therefore for 99's divisibility by 9 and 3.

Uses, culture, and notable facts

Beyond pure mathematics, 99 appears frequently in culture and commerce: famous songs and chants use the number (for example the traditional song often called "99 Bottles of Beer"), and consumer prices are commonly set just below a round number at .99. In recreational mathematics it is a simple, illustrative example for factorization, repeating decimals, and parity. For further reading and related entries see odd numbers and other basic integer topics via reference.

Ninety-nine's simple structure (3^2 × 11), its place as 10^2 − 1, and its cultural visibility make it a useful and familiar example in teaching arithmetic, divisibility rules, and elementary number theory.