What's the speciality of the number '5040'?

In mathematics there are many interesting number like  π,  The golden ratio φ , 1729 , and many others

In such interesting numbers 5040 is also in the list of them. So this number 5040 looks normal like without any special characteristics but it do posses some of the very eye catching characteristics.

So if we start from just looking at the number we can say that it is  7! that is 5040 and if we closely look at this in the view of permutation then we can see that it is the number of permutation of 4 items out of 10 choices that is \(^{10}C_4\)=10*9*8*7

Now if we look at this number in a mathematical and analytical way

1. We can see that 5040 has exactly 60 factors including 1 and 60 itself which are 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.
2. Again this number 5040 is unique in another way that is , this number is the sum of 42 primes that is 42 consecutive primes and that is 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 +163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229.
3. 5040 this number is just one less than a square, making (7, 71) a Brown number pair that is 7!+1=5041 which is 1 less than 5040. Now in mathematics a brown number are pairs of the numbers taken like (n,m)  that solve the Brocard's Problem {Brocard’s problem in mathematics is where we have to find an integer values of m and n mentioned above for which n!+1=\(m^2\) } values of  and that solve Brocard’s problem are called Brown Numbers . Until 2019, we found the existence of three brown number pairs which are (4,5) , (5,11) and (7,71)
4. Our number 5040 is also a superior highly composite number. Now in mathematics a highly composite number is a number which has more divisors than any other number which is smaller than the number itself and also there prime factorisation should be the multiplication of consecutive primes.

• Checking

5040=\(2^4\)*\(3^2\)*5*7 

• So it has the product of consecutive primes
• 5040 has more number of divisors (5040 has 60 divisors) than any other number which is smaller than 5040.

So it is a highly composite number.

5. This number 5040 is also a colossally abundant number. The first 15 colossally abundant numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800.

Some small but interesting facts about 5040

1. 5040 can be divided by every number from 1 to 12 only exception to this is 11.
2. However 5038, the closest number to 5040 can be divided by 11.
3. The 12th part of the number 5040 is divisible by 12.

5040 in our daily life

• Double of this number(that is 5040) is equal to the number of minutes in a week that is
  • In a week we have 24*7=168 hours now 168*60 hours = 10080  minuets . So we have 10080 minuets in a week. Now if we see that 5040*2=10080 just amazing.
• Now the radius of moon is 1079.3839 miles and the radius of the Earth is 3958.756 miles so adding the two we get 5038.1399 miles which is 99.96% accurate to 5040.

Okay now if we look into history and I will say some philosophy we can see that.

• In the record of Plato’s laws 5040 is a convenient number that can be used for dividing a number of things including citizens or a city state.