30 is a natural number that comes after 29 and before 31. It is an even, composite integer and the product of the first three prime numbers: 2 × 3 × 5. Because it is the product of three distinct primes, 30 is the smallest sphenic number and also the third primorial (2·3·5). Its Roman numeral notation is XXX.
Mathematical properties
The positive divisors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. It has eight divisors in total. The sum of its proper divisors is 42, which makes 30 an abundant number (the sum exceeds the number itself). Euler's totient function gives φ(30) = 8, meaning eight integers less than 30 are coprime to it.
In numeral systems 30 can be represented as 11110 in binary, 36 in octal, and 1E in hexadecimal. As a Harshad number in base ten, 30 is divisible by the sum of its digits (3 + 0 = 3). It appears frequently as a useful modulus and factor in elementary number theory because of its small prime factorization.
Language and history
The English word "thirty" derives from Old English þrītig ("three tens"); many Germanic languages show a related form. Historically, groups and counting systems often used tens as basic units, so a name equivalent to "three tens" is widespread.
Uses and cultural references
Thirty appears in many everyday contexts: a typical number of days in some months, an age milestone often associated with adulthood, and a score in sports such as tennis (15, 30, 40). In chemistry, 30 is the atomic number of zinc. It also features in calendars, legal time periods, and measures where round multiples of ten are convenient.
Because of its simple prime factors and moderate size, 30 serves as a common example in teaching divisibility, least common multiples, and basic factorization. Mathematicians and educators use it to illustrate properties like sphenic numbers, primorials, abundance, and totients without large computations.