1338 can refer either to the integer that follows 1337 or to the calendar year 1338 AD. As a subject it is treated both mathematically and historically: the number has a simple prime factor structure and a handful of canonical representations, while the year sits in the early high‑middle‑ages, a time of political conflict and social change across Europe and Asia.
Mathematical properties
As an integer, 1338 is composite. Its prime factorization is 2 × 3 × 223. It has eight positive divisors in total: 1, 2, 3, 6, 223, 446, 669 and 1338. The sum of its proper divisors is 1,350, which exceeds 1338, so the number is classified as abundant. Arithmetic functions give small tidy values: it has 8 divisors and Euler's totient function φ(1338) = 444.
- Roman numerals: MCCCXXXVIII
- Binary: 10100111010
- Hexadecimal: 0x53A
- Not a perfect square or cube (√1338 ≈ 36.58)
1338 as a year (AD)
The year 1338 falls in the first decades of the 14th century, a period marked by dynastic conflict, regional wars and shifting alliances. It lies early in the long series of hostilities later called the Hundred Years' War (commonly dated from 1337), and during an era when medieval polities in Europe, the British Isles, and parts of Asia were experiencing political restructuring and military campaigning.
Although detailed lists of local occurrences vary by region, the general backdrop for the year includes intensifying competition among western European monarchies, commercial and naval rivalries in the Low Countries and the English Channel, and continuing dynastic struggles in places such as the Holy Roman Empire and the Italian city‑states. In East Asia, the mid‑14th century was also a time of contested authority and realignments that would shape later medieval states.
Significance and notable facts
As a number, 1338 is a straightforward example used to illustrate factorization, divisor sums and classification (abundant versus deficient numbers). As a year, it is representative of the turbulent mid‑14th century: not famous for a single globally decisive event, it belongs to a span that produced major long‑term consequences such as protracted warfare, shifting trade networks, and the social stresses that culminated later in the century.