77 (number): properties, representations, and uses
An overview of the integer 77: arithmetic properties, factorizations, numerical representations, and notable occurrences such as its role as the atomic number of iridium.
77 is an integer that follows 76 and precedes 78. In the usual counting numbers it is a natural number. Written in base ten it is a two-digit repdigit — the same digit repeated — and thus a simple palindrome (77 reads the same forwards and backwards). It appears in elementary lists of composite numbers and in many elementary arithmetic examples.
Basic arithmetic properties
As a composite number, 77 factors into the product of two primes: 7 and 11, so 77 = 7 × 11. Because it is the product of two (distinct) primes it is a semiprime. Its positive divisors are 1, 7, 11 and 77, giving a divisor count of 4. The sum of all divisors is 96, and the sum of proper divisors (1 + 7 + 11 = 19) is less than 77, so 77 is a deficient number.
Number-theoretic functions
Standard arithmetic functions return simple values for 77: Euler's totient function φ(77) = 60, since φ(pq) = (p−1)(q−1) for distinct primes p and q. The Möbius function μ(77) = 1 because 77 is squarefree with an even number of prime factors. The integer is also a Blum integer, because its two prime factors (7 and 11) are both congruent to 3 modulo 4.
Representations and patterns
Besides base ten, 77 has compact representations in other common bases: binary 1001101, octal 115, and hexadecimal 4D. As a two-digit repdigit in base ten, it matches the general identity that any double-digit repdigit NN equals 11×N (so 77 = 11×7). Such repdigits are elementary examples in place-value arithmetic and help illustrate divisibility by 11.
Occurrences and context
Outside pure arithmetic, 77 appears as the atomic number of the chemical element iridium (Ir), a dense, corrosion-resistant transition metal used in high-temperature applications and in certain alloys. The number also shows up in cultural and recreational contexts — as an emblematic two-digit palindrome and in identifiers — but its mathematical traits (semiprime, palindromic repdigit, Blum integer) are the most systematically studied.
- Prime factorization: 7 × 11
- Divisors: 1, 7, 11, 77
- Euler totient φ(77) = 60
- Sum of proper divisors = 19 (deficient)
For further reading on basic integer properties and palindromic numbers see general references on elementary number theory and digit patterns; introductory resources and databases collect related facts and sequences. More technical discussions of semiprimes and Blum integers appear in algebraic and cryptanalytic literature. Related numeric entries and succession
Author
AlegsaOnline.com 77 (number): properties, representations, and uses Leandro Alegsa
URL: https://en.alegsaonline.com/art/133951