Fifteen (number)

Mathematical properties

As a composite number 15 has four positive divisors: 1, 3, 5 and 15. The sum of its proper divisors is 1 + 3 + 5 = 9, so 15 is a deficient number. It is the product of two distinct primes (a semiprime), and its Euler totient value is φ(15) = 8. Notable combinatorial identities involving 15 include binomial coefficients: 15 = C(6,2) = C(6,4). It is the fifth triangular number (1+2+3+4+5 = 15) and also the third hexagonal number.

Representations and notation

• Decimal: 15
• Binary: 1111
• Octal: 17
• Hexadecimal: F
• Roman numerals: XV

Because the digit sum 1+5 = 6 is divisible by 3, divisibility tests show 15 is divisible by 3 and by 5. For a brief introduction to its arithmetic properties see basic number references, and for modular facts consult general resources such as modular arithmetic guides.

Cultural, scientific and practical contexts

Fifteen occurs in many everyday and cultural settings. In sports, a rugby union team fields 15 players. Tennis scoring traditionally advances 0, 15, 30, 40. In chemistry, the atomic number 15 identifies the element phosphorus. Many societies mark the fifteenth birthday with specific rites of passage, famously the Latin American quinceañera.

Examples and notable references

A famous puzzle named the "15-puzzle" is a sliding-tile game on a 4×4 grid that became popular in the 19th century. In recreational mathematics 15 appears frequently because it is small enough to enumerate fully yet large enough to illustrate patterns in divisors, polygonal numbers and combinations. For further reading on puzzles and recreational uses see recreational mathematics resources and overviews at general science sites (Roman numerals and notation).