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: $738$, $741$, $747$, $771$, $878$
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): $5040$