25 (number)

Overview of the integer 25: mathematical properties, notation and representations, historical and cultural uses, examples and notable distinctions.

Overview

25 is the natural number after 24 and before 26. It is an odd composite integer and the square of the prime number 5. As a square, it is often introduced early in arithmetic and appears in many elementary mathematical contexts.

Mathematical properties

Key numeric facts about 25 include its prime factorization 5², and its positive divisors {1, 5, 25}. The sum of proper divisors is 6, which makes 25 a deficient number. Important arithmetic values are φ(25)=20 (Euler's totient function) and σ(25)=31 (sum of divisors).

Square: 25 = 5², the fifth perfect square (see square number).
Odd: it is not divisible by 2 (see odd number).
Base representations: binary 11001, octal 31, hexadecimal 19, Roman numerals XXV.
Sum of first five odd numbers: 1 + 3 + 5 + 7 + 9 = 25, illustrating a general pattern for square numbers.

Notation and history

The symbol for the quantity 25 depends on the numeral system: in modern Western usage it is written '25' in Hindu–Arabic numerals and 'XXV' in Roman numerals. Historical references to the number appear in accounting, calendaring and measurement records across many cultures whenever counting beyond the low tens was required.

Uses and examples

25 appears in everyday contexts: a quarter of 100; the US quarter coin equals 25 cents; a 25th anniversary is commonly called a silver anniversary. In science and technology, the atomic number 25 corresponds to manganese, and in statistics or sampling a group of 25 items is a convenient small sample size.

Notable distinctions

As a square of a prime, 25 is a prime power and displays properties typical of such numbers. It is the fifth square, the product of two-digit decimal digits 2 and 5 equals 10, and it often serves as an elementary example when teaching exponents, divisibility, modular arithmetic or geometric area (a 5 by 5 square has area 25).