1144 (number and year)

Mathematical properties

In prime factor form, 1144 = 2^3 × 11 × 13. From this decomposition one can derive several facts: it has (3+1)(1+1)(1+1) = 16 positive divisors, and the Euler totient φ(1144) equals 480. The sum of all positive divisors σ(1144) is 2520, so the sum of proper divisors (σ − 1144) is 1376. Because the proper-divisor sum exceeds the number itself, 1144 is classified as an abundant number.

• Prime factorization: 2^3 · 11 · 13
• Divisors (16): 1, 2, 4, 8, 11, 13, 22, 26, 44, 52, 88, 104, 143, 286, 572, 1144
• Sum of divisors σ(1144) = 2520 (notably equal to lcm(1,2,...,10))
• Euler totient φ(1144) = 480
• Abundant: proper-divisor sum = 1376 > 1144

Representations and notation

1144 is even and divisible by 8, so it is a multiple of a small power of two. Common positional representations include binary 10001111000, octal 2170, and hexadecimal 478. In Roman numerals it is written MCXLIV. Its digital sum is 1+1+4+4 = 10, so it is not divisible by its digit sum (not a Harshad number in base 10).

Historical and contextual notes

When used as a year number, AD 1144 belongs to the 12th century. One widely cited event of AD 1144 is the fall of the city of Edessa to the atabeg Zengi, an outcome that alarmed the Latin states in the Levant and contributed to calls for a large new military expedition in the West (the Second Crusade). As with most integers, 1144 appears in catalogues, serial designations and numbering schemes across science and administration, but its best-known non-mathematical association is as the date marker for events of the mid-12th century.

Summary: 1144 is a composite, abundant integer with eight small power-of-two factors in its makeup and several easily computed arithmetic functions; it also serves as a concise chronological label for historical study of the 1100s.