5040 (number)

← 5039

5040

5041 →

List of numbers — Integers

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Cardinal

five thousand forty

Ordinal

5040th
(five thousand fortieth)

Factorization

2⁴× 3²× 5 × 7

Divisors

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

VXL

1001110110000₂

20220200₃

1032300₄

130130₅

35200₆

11660₈

2B00₁₂

13B0₁₆

CC0₂₀

3W0₃₆

5040 is a factorial (7!), a superior highly composite number, a colossally abundant number, and the number of permutations of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040).

Plato mentions in his Laws that 5040 is a convenient number to use for dividing many things (including both the citizens and the land of a state) into lesser parts. He remarks that this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11 (however, it is not the smallest number to have this property; 2520 is). He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number 5038, which is divisible by 11. Plato also took notice of the fact that 5040 can be divided by 12 twice over. Indeed, Plato's repeated insistence on the use of 5040 for various state purposes is so evident that it is written, "Plato, writing under Pythagorean influences, seems really to have supposed that the well-being of the city depended almost as much on the number 5040 as on justice and moderation."

Jean-Pierre Kahane has suggested that Plato's use of the number 5040 marks the first appearance of the concept of a highly composite number, a number with more divisors than any smaller number.

If $\sigma (n)$ is the divisor function and $\gamma$ is the Euler–Mascheroni constant, then 5040 is the largest of the known numbers (sequence in the OEIS) for which this inequality holds:

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