1035: the integer and the year AD 1035

Mathematical properties

The prime factorization of 1035 is 3^2 × 5 × 23. From this follow basic arithmetic invariants: it has 12 positive divisors and the sum of all divisors equals 1872, so the sum of proper divisors is 837—less than 1035—making it a deficient number. 1035 is a triangular number: it equals 45×46/2, the 45th triangular number. In base 10 its digit sum is 9, and because 1035 is divisible by 9 it is a Harshad (Niven) number.

Common numeral representations include binary 10000001011, hexadecimal 0x40B, and Roman numerals MXXXV. These forms are useful in computer science and historical notation respectively.

Divisors and small facts

• Prime factors: 3, 5, 23 (with 3 squared)
• Number of positive divisors: 12
• Sum of divisors (σ): 1872; sum of proper divisors: 837
• Triangular: T45 = 1035
• Harshad number in base 10

Year AD 1035: historical context

The year 1035 is notable in medieval North Sea politics. King Cnut (Canute) the Great, who had ruled over England, Denmark and Norway in various combinations, died in 1035. His death triggered succession disputes: his sons Harthacnut and Harold (Harold Harefoot) were rival claimants in England and Denmark, producing a period of instability. In Normandy the death of Duke Robert left his illegitimate son William as a child duke; this circumstance set the stage for later developments in Norman and English history.

These events influenced dynastic alignments and the balance of power in northern Europe for the remainder of the 11th century, contributing to the political background of the later Norman conquest of England and to shifting relationships among Scandinavian and English rulers.