408 is both an integer and a calendar year. In the Julian calendar it was a leap year (divisible by four). Historically it falls in the early fifth century, a moment of intensified political and military pressure on the western Roman Empire. As an integer, 408 is an even composite with the prime factorization 2^3·3·17 and several elementary arithmetic properties that make it a routine example in number theory.
Year 408 (AD): context and events
The year witnessed major shifts in western Roman politics and military command. The magister militum Flavius Stilicho, long the dominant military figure in the West and a guardian of the young emperor Honorius, was removed from power and executed after court intrigues and factional violence. His fall precipitated purges of his supporters and unsettled relationships with federate troops and barbarian allies.
Taking advantage of Roman weakness, the Visigoths under their leader Alaric renewed pressure on Italy. Movements of Gothic forces during and after 408 intensified the threat to Rome itself and contributed to a sequence of events that culminated in the sack of Rome a few years later. More broadly, the removal of central military authority aggravated frontier insecurity in Gaul, Hispania and along the Danube, encouraging migration and local usurpations.
The number 408: arithmetic properties
As a number, 408 = 2^3 × 3 × 17. It has 16 positive divisors and the sum of all divisors equals 1,080, so the sum of proper divisors is 672, greater than 408; therefore it is classified as an abundant number. Its Euler totient φ(408) equals 128, and the sum of its decimal digits (4 + 0 + 8 = 12) divides the number, making 408 a Harshad number in base ten. In Roman numerals it is written CDVIII.
Significance
In historical surveys 408 is remembered less for a single decisive battle than for the political rupture that weakened Western command and the renewed incursions of Gothic groups. In arithmetic, 408 is representative of modestly structured composite integers that illustrate divisor sums, multiplicative functions and simple divisibility tests. Together the year and the integer demonstrate how a single label can connect distinct domains of calendar, politics and mathematics.