Overview
Forty-five is the natural number following 44 and preceding 46. As an integer it is composite, with the prime factorization 3^2 · 5. It is classified as an odd number and is divisible by 1, 3, 5, 9, 15 and 45. Basic arithmetic properties such as its divisor count and totient function reflect this factorization.
Mathematical properties
Important arithmetic characteristics include: it has six positive divisors, so d(45)=6; the sum of its divisors is 78, σ(45)=78; it is a deficient number because the sum of proper divisors (33) is less than 45; and Euler's totient φ(45)=24. In combinatorics 45 appears as a binomial coefficient: C(10,2)=45, the number of unordered pairs from ten elements.
- Prime factorization: 3^2 · 5
- Divisors: 1, 3, 5, 9, 15, 45
- Euler totient: 24
- Sum of divisors: 78
Figurate and representational aspects
Forty-five is the 9th triangular number (9·10/2=45), meaning it can be arranged as an equilateral triangular array of points. It is also the 5th hexagonal number, found from the formula n(2n−1) for n=5. In base representations: binary 101101, octal 55, hexadecimal 2D, and in Roman numerals XLV. Because its decimal digit sum is 9 and 45 is divisible by 9, it is a Harshad number in base 10.
Applications, examples and notable appearances
Number 45 appears in everyday contexts and specialized fields: 45° is a commonly used angle in geometry and design; the 45 rpm record format (a standard for single records) and the .45 caliber cartridge in firearms are named using this numeral. In elementary combinatorics the value C(10,2)=45 counts pairs from ten items, a simple example of its occurrence in counting problems. For concise technical summaries and further reading see additional notes, detailed properties at mathematical references, or combinatorial examples at related resources.