Determine the numerical properties of 72

We start by listing out divisors for 72

Divisor | Divisor Math |
---|---|

1 | 72 ÷ 1 = 72 |

2 | 72 ÷ 2 = 36 |

3 | 72 ÷ 3 = 24 |

4 | 72 ÷ 4 = 18 |

6 | 72 ÷ 6 = 12 |

8 | 72 ÷ 8 = 9 |

9 | 72 ÷ 9 = 8 |

12 | 72 ÷ 12 = 6 |

18 | 72 ÷ 18 = 4 |

24 | 72 ÷ 24 = 3 |

36 | 72 ÷ 36 = 2 |

Since 72 has divisors

A perfect number is a number who has a divisor sum equal to itself. An abundant number is a number who has a divisor sum greater than the number, otherwise, it is deficient.

Divisor Sum = 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36

Divisor Sum = 123

Since our divisor sum of 123 > 72, 72 is

A number is even if it is divisible by 2, else it is odd

36 = | 72 |

2 |

Since 36 is an integer, 72 is divisible by 2, and therefore, it is an

This can be written as A(72) = Even

A number is evil is there are an even number of 1's in the binary expansion, else, it is odious.

Using our decimal to binary calculator, we see the binary expansion of 72 is 1001000

Since there are 2 1's in the binary expansion which is an even number, 72 is an

A number is triangular if it can be stacked in a pyramid with each row above containing one item less than the row before it, ending with 1 item at the top

Using a bottom row of 12 items, we cannot form a pyramid with our numbers, therefore 72 is

Triangular number: 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

A number n is rectangular is if there is an integer m such that n = m(m + 1)

The integer m = 8 satisifes our rectangular number property, since 8(8 + 1) = 72

Rectangular number: 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

A number (n) is automorphic (curious) if n

72

Since 5184 does not end with 72, it is

A number (n) is undulating if the digits that comprise n alternate in the form abab

Since 72 < 100, we only perform the test on 3 digit numbers or higher

A number (n) is a square if there exists a number m such that m

Analyzing squares, we see that 8

Therefore, 72 is

A number (n) is a cube if there exists a number m such that m

Analyzing cubes, we see that 4

Therefore, 72 is

A number (n) is a palindrome if the number read backwards equals the number itself

The number read backwards is 27

Since 72 <> 27, it is

A number is a palindromic prime if it is both prime and a palindrome

From above, since 72 is not both prime and a palindrome, it is

A number is repunit if every digit is equal to 1

Since there is at least one digit in 72 not equal to 1, then it is

A number (n) is apocalyptic power if 2

2

Since 2

A pentagonal number is one which satisfies the form:

n(3n - 1) | |

2 |

Using n = 8, we have:

8(3(8 - 1) | |

2 |

8(24 - 1) | |

2 |

8(23) | |

2 |

184 | |

2 |

92 ← Since this does not equal 72, this is

Using n = 7, we have:

7(3(7 - 1) | |

2 |

7(21 - 1) | |

2 |

7(20) | |

2 |

140 | |

2 |

70 ← Since this does not equal 72, this is

Pentagonal number: 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

A number n is hexagonal is if there is an integer m such that n = m(2m - 1)

No integer m exists such that m(2m - 1) = 72, therefore 72 is not hexagonal

Hexagonal number: 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

A number n is heptagonal is if there is an integer m such that:

m = | n(5n - 3) |

2 |

No integer m exists such that m(5m - 3)/2 = 72, therefore 72 is not heptagonal

Heptagonal number: 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

A number n is octagonal is if there is an integer m such that n = m(3m - 3)

No integer m exists such that m(3m - 2) = 72, therefore 72 is not octagonal

Octagonal number: 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

A number n is nonagonal is if there is an integer m such that:

m = | n(7n - 5) |

2 |

No integer m exists such that m(7m - 5)/2 = 72, therefore 72 is not nonagonal

Nonagonal number: 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

A tetrahedral (Pyramidal) number is one which satisfies the form:

n(n + 1)(n + 2) | |

6 |

Using n = 7, we have:

7(7 + 1)(7 + 2) | |

6 |

7(8)(9) | |

6 |

504 | |

6 |

84 ← Since this does not equal 72, this is

Using n = 6, we have:

6(6 + 1)(6 + 2) | |

6 |

6(7)(8) | |

6 |

336 | |

6 |

56 ← Since this does not equal 72, this is

An m digit number n is narcissistic if it is equal to the square sum of it's m-th powers of its digits

72 is a 2 digit number, so m = 2

Square sum of digits

Square sum of digits

Square sum of digits

Since 53 <> 72, 72 is

The n

C_{n} = | 2n! |

(n + 1)!n! |

Using n = 6, we have:

C_{6} = | (2 x 6)! |

6!(6 + 1)! |

Using our factorial lesson to evaluate, we get

C_{6} = | 12! |

6!7! |

C_{6} = | 479001600 |

(720)(5040) |

C_{6} = | 479001600 |

3628800 |

C

Since this does not equal 72, this is

Using n = 5, we have:

C_{5} = | (2 x 5)! |

5!(5 + 1)! |

Using our factorial lesson to evaluate, we get

C_{5} = | 10! |

5!6! |

C_{5} = | 3628800 |

(120)(720) |

C_{5} = | 3628800 |

86400 |

C

Since this does not equal 72, this is

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