5,040
5,040 is 7 factorial, and the population Plato proposed for an ideal city.
- Even
- Composite
Where 5,040 fits
Why 5,040 is special
Plato chose 5,040 in the Laws for a practical reason: it divides evenly in 59 different ways, including by every number from 1 to 10 except 7's neighbours, so land, taxes and military duties could be apportioned without fractions. It is also 7!, the number of ways to arrange seven objects, which is why it appears wherever orderings are counted.
Factors & digit properties
- Divisors (60)
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040
- Prime factorisation
- 2^4 × 3^2 × 5 × 7
- Digit sum
- 9
- Digital root
- 9
Representations
- Decimal
- 5040
- Binary
- 1001110110000
- Octal
- 11660
- Hexadecimal
- 13b0
- Scientific notation
- 5.040 × 10^3
Other mathematical patterns
- Abundant number
- Factorial number
- Highly composite number
- Harshad (Niven) number
Related numbers & connections
40,320 is 8! and exactly eight times 5,040. Each step in the factorial sequence multiplies by the next integer, which is why the values pull away from anything polynomial so quickly.
View 40320 →Consecutive factorials, 6! and 7!. Each is the number of ways to arrange that many distinct objects, and the jump between them — 720 to 5,040 — is the point where factorial growth starts to feel unreasonable.
View 720 →History
In the Laws, Plato proposes 5,040 households because the number divides evenly 59 different ways, including by every integer from 1 to 12 except 11. That let land, taxes and military service be apportioned without fractions — a design choice about administration rather than a mystical claim about the number itself.
DocumentedHistoryCulture: classical Greek philosophy
Common questions
Is 5,040 a prime number?
No, 5,040 is not prime.
What are the factors of 5,040?
The factors of 5,040 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040.
About this page's data
Mathematical facts:computed directly from the number's value — nothing to source or verify externally.
Contextual records: 1 reviewer-confirmed record, sourced and reviewed before publication.
Last reviewed: 3 August 2026