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