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
- Scientific notation
- 5.040 × 10^3
Roman numerals
Roman numerals as modelled here goes up to 3,999.
Binary
1 0 0 1 1 1 0 1 1 0 0 0 0
1 × 4,096 + 0 × 2,048 + 0 × 1,024 + 1 × 512 + 1 × 256 + 1 × 128 + 0 × 64 + 1 × 32 + 1 × 16 + 0 × 8 + 0 × 4 + 0 × 2 + 0 × 1 = 5,040
Octal
1 1 6 6 0
1 × 4,096 + 1 × 512 + 6 × 64 + 6 × 8 + 0 × 1 = 5,040
Hexadecimal
1 3 b 0
1 × 4,096 + 3 × 256 + b × 16 + 0 × 1 = 5,040
Read as powers of two: 4,096 + 512 + 256 + 128 + 32 + 16 = 5,040.
In historical numeral systems
Egyptian
5 lotuses, 4 hobbles
5 × 1,000 + 4 × 10 = 5,040. The signs are simply repeated and added; where they sit on the line is a matter of layout, not value.
Systems that simply cannot reach this value are left out. Where a script needs a font most readers do not have, the reading is given instead of the glyphs — see number systems for each system in full.
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
Mathematics
In 1984 Guy Robin proved that the Riemann hypothesis is true if and only if the sum of the divisors of n stays below e^γ · n · ln ln n for every n greater than 5,040. The bound fails at exactly the highly composite numbers up to and including 5,040 and, if the hypothesis holds, never again. So one of the central open problems in mathematics has a precise threshold, and the threshold is 7 factorial.
DocumentedMathematicsPeriod: 19845,040 = 7! is the number of ways to arrange seven distinct objects in order, which is why it appears wherever orderings are counted: seven people in a row, seven cards dealt in sequence, seven tasks scheduled. The growth is the point — 7! is 5,040 while 10! is over three and a half million — and it is the reason brute-force search over permutations stops being practical almost immediately.
DocumentedMathematics
Common questions
Why did Plato propose 5040 households?
For administration, not mysticism. In the Laws he chose it because it divides evenly 59 different ways, including by every integer from 1 to 12 except 11, so land, taxes and military service could be apportioned without fractions. It is also 7 factorial — the number of ways to arrange seven objects.
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: 3 reviewer-confirmed records, sourced and reviewed before publication.
Last reviewed: 15 July 2026