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 $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$, $1680$, $2520$:
$\map \tau {5040} = 60$


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


Arithmetic Functions on $5040$

\(\displaystyle \map \tau { 5040 }\) \(=\) \(\displaystyle 60\) $\tau$ of $5040$
\(\displaystyle \map \sigma { 5040 }\) \(=\) \(\displaystyle 19 \, 344\) $\sigma$ of $5040$


Also see



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.


Sources