Overview

1173 is a positive integer that lies between 1172 and 1174. In arithmetic it is an odd composite number with several elementary and noteworthy properties. Its prime factorization is 3 × 17 × 23, so it is the product of three distinct primes.

Arithmetic properties

Because 1173 = 3 × 17 × 23 it has 2³ = 8 positive divisors. The complete list of divisors is: 1, 3, 17, 23, 51, 69, 391, 1173. The sum of all divisors σ(1173) equals 1728, a familiar cube (12³), while the sum of proper divisors is 555, which is less than 1173; therefore 1173 is classified as a deficient number. Euler's totient function gives φ(1173) = 704.

Representations and notation

1173 can be written in several positional and symbolic notations: in binary it is 10010010101, in hexadecimal it is 0x495, and in Roman numerals it is MCLXXIII. As an odd integer it can be expressed as a sum of consecutive integers in more than one way; examples of positive consecutive sums equal to 1173 include:

  • 1173 (one term)
  • 586 + 587 (two terms)
  • 390 + 391 + 392 (three terms)
  • 61 through 77 (seventeen terms)
  • 40 through 62 (twenty-three terms)

Notable numeric facts

Because its prime factors are distinct, 1173 is a sphenic number (product of three distinct primes). The coincidence σ(1173) = 1728 is sometimes remarked upon because 1728 = 12³ gives a tidy connection between the divisor function and small integer powers. Its aliquot sum (sum of proper divisors) starts the short sequence 1173 → 555 → 357 → ..., showing it is not amicable or perfect.

Year 1173 (AD)

When taken as a calendar year, 1173 AD (MCLXXIII) is remembered in architecture and medieval history: for example, construction of the bell tower at Pisa—the structure now known as the Leaning Tower of Pisa—began in 1173. References to the year are common in chronologies of European medieval events.

Summary

1173 combines simple but interesting arithmetic features: an elementary three‑prime factorization, a small set of divisors, a totient of 704, and a divisor sum equal to 1728. It serves as both a number with neat numeric coincidences and as a label for historical events from the late 12th century.