Overview
The integer 78 is the natural number that follows 77 and precedes 79. It is an even composite number with several familiar arithmetic and cultural roles. In elementary counting it is the sum of the first twelve positive integers and appears frequently in combinatorics and common counts.
Mathematical properties
In prime factorization form 78 = 2 × 3 × 13. Its complete list of positive divisors is:
- 1, 2, 3, 6, 13, 26, 39, 78
The sum of all divisors is 168, so the sum of proper divisors (excluding 78 itself) is 90. Because 90 > 78 the number is classified as abundant; moreover it is semiperfect because 39 + 26 + 13 = 78. The count of divisors is eight, the Möbius function μ(78) = −1 (three distinct prime factors), and the Euler totient φ(78) = 24.
Combinatorial and figurate roles
78 is the 12th triangular number: 1 + 2 + ... + 12 = 78. Equivalently, it equals the binomial coefficient C(13,2), the number of unordered pairs chosen from 13 items. These appearances connect 78 to simple combinatorial counts and to figurate-number sequences.
History, science and culture
Outside pure arithmetic, 78 is notable as the atomic number of platinum, an important transition metal (see atomic number 78). In astronomy Messier 78 is a well-known reflection nebula in Orion. In cultural contexts, a standard tarot deck contains 78 cards, and early phonograph records commonly spun at 78 revolutions per minute, giving rise to the term "78" for that format.
Notable distinctions
Compared with nearby integers, 78 stands out as both triangular and abundant. It is not prime, not a perfect square, and is semiperfect by virtue of a subset of its divisors summing to it. For broader definitions and related concepts see general articles on the nature of numbers and on classifications such as composite and abundant numbers.