# 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 $5040$

## 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: $738$, $741$, $747$, $771$, $878$

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