1105 (number and year)

Mathematical properties

As an integer 1105 factors as 5 × 13 × 17. Its positive divisors are 1, 5, 13, 17, 65, 85, 221 and 1105. The sum of all divisors is 1,512, so the sum of proper divisors (407) is smaller than the number, making 1105 a deficient number.

Several arithmetic functions give compact values: Euler's totient φ(1105) = 768, because φ(n) multiplies n by (1 − 1/p) for each prime factor. The Carmichael function λ(1105), the exponent of the multiplicative group modulo 1105, equals lcm(4,12,16) = 48.

1105 is notable in primality testing because it is a Carmichael number. Carmichael numbers are composite integers that nevertheless satisfy a^n−1 ≡ 1 (mod n) for all integers a coprime to n, so they can fool simple Fermat primality tests. Korselt's criterion explains this: a Carmichael number must be squarefree and for every prime p dividing n, p−1 must divide n−1; 1105 satisfies these conditions.

1105 as a calendar year (AD)

The year AD 1105 falls in the High Middle Ages, a period of political consolidation, monastic and scholastic activity, and cross‑Mediterranean contact that followed the First Crusade. Across Eurasia established dynasties and emergent states—such as the Norman realms in western Europe, the Song dynasty in China and various Seljuk and regional polities in the Islamic world—shaped politics, trade and culture during this era.

1105 BC and broader chronology

The notation 1105 BC places the date in the late second millennium BCE, a time of local transitions after the widespread disruptions conventionally associated with the Late Bronze Age collapse. Different regions experienced different rhythms of recovery and change: political realignments, migration, and technological developments such as advances in ironworking were important themes across many areas.

Uses, references and notable facts

• Practical: the number 1105 appears routinely as an identifier—street numbers, product or model codes, legal or administrative entries—without special mathematical significance in those contexts.
• Mathematical note: 1105 is one of the early Carmichael numbers (after 561); its behaviour under simple primality tests makes it a standard example in number theory and computational mathematics.
• Arithmetic summary: factorization 5·13·17, φ(1105)=768, λ(1105)=48, proper divisors sum to 407 (deficient).