# 5040

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## Number

$5040$ (**five thousand and forty**) is:

- $2^4 \times 3^2 \times 5 \times 7$

- The product of consecutive integers in $2$ different ways:
- $5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$

- The $7$th factorial after $1$, $2$, $6$, $24$, $120$, $720$:
- $5040 = 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

- The $19$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$, $1680$, $2520$:
- $\map {\sigma_0} {5040} = 60$

- The $19$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$, $1680$, $2520$:
- $\dfrac {\map {\sigma_1} {5040} } {5040} = \dfrac {19 \, 344} {5040} \approx 3 \cdotp 838$

### Arithmetic Functions on $5040$

\(\ds \map {\sigma_0} { 5040 }\) | \(=\) | \(\ds 60\) | $\sigma_0$ of $5040$ | |||||||||||

\(\ds \map {\sigma_1} { 5040 }\) | \(=\) | \(\ds 19 \, 344\) | $\sigma_1$ of $5040$ |

## Also see

*Previous ... Next*: Factorial*Previous ... Next*: Highly Composite Number*Previous ... Next*: Superabundant Number

## Historical Note

The philosopher Plato decided that the exact number of citizens suitable for his ideal city was $5040$.

His reasons included:

- $5040$ has $59$ divisors excluding itself
- Can be divided by all numbers from $1$ to $10$ and so can be assembled for various wartime or peacetime collective activities into so many equal teams
- Subtracting two hearths (that is, people) from the total, you get $5038$, which is divisible by $11$ as well.

- -- Plato's
*Laws*: $738$, $741$, $747$, $771$, $878$

- -- Plato's

In the science of campanology, a complete sequence of Stedman triples contains $5040$ changes, and takes between $3$ and $4$ hours to accomplish.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5040$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5040$