60 (number)

An overview of the integer 60: its mathematical properties, factorization and divisors, role in measurement systems and timekeeping, historical use of base‑60, and notable cultural facts.

60 is the integer that follows 59 and precedes 61. It is an even, composite number commonly written as "sixty" in English and in Roman numerals as LX. In common positional notation it appears as 60 (decimal), 111100 (binary) and 3C (hexadecimal). The atomic number 60 identifies the element neodymium in the periodic table. For a general reference, see 60.

Mathematical properties

Sixty has the prime factorization 2^2 × 3 × 5. That factorization gives it a relatively large number of divisors for its size, making it a commonly used example in elementary number theory. It is the least common multiple of the integers 1 through 6, so any integer divisible by 1..6 is divisible by 60. This LCM property underlies many practical applications; for more on divisibility, see divisibility rules.

Key invariants: Euler's totient function φ(60)=16, and the sum of all divisors of 60 is 168, which makes 60 an abundant number (sum of proper divisors exceeds the number itself). Because of its many divisors, 60 is also classified among highly composite numbers used in divisor‑related problems.

Divisors

Positive divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
Proper divisors (exclude 60): 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30.

These divisors make 60 easy to split into many equal parts: halves, thirds, quarters, fifths, sixths, tenths, twelfths and so on.

History and practical uses

One of the most lasting legacies of 60 is the sexagesimal (base‑60) numeral system developed in ancient Mesopotamia. Elements of that system survive in modern measurements: 60 seconds make a minute, 60 minutes make an hour, and there are 360 (=6×60) degrees in a circle. These conventions persist in timekeeping, navigation, astronomy and geometry because 60's many factors simplify fractional subdivisions.

Everyday examples and cultural notes: an hour has 60 minutes and a minute has 60 seconds; the 60th anniversary is widely celebrated as a major milestone in many traditions. For a concise list of divisors and related facts, consult divisors and properties of 60.