 # Numerical properties of 72

## Enter Integer

Show numerical properties of 72

We start by listing out divisors for 72
DivisorDivisor Math
172 ÷ 1 = 72
272 ÷ 2 = 36
372 ÷ 3 = 24
472 ÷ 4 = 18
672 ÷ 6 = 12
872 ÷ 8 = 9
972 ÷ 9 = 8
1272 ÷ 12 = 6
1872 ÷ 18 = 4
2472 ÷ 24 = 3
3672 ÷ 36 = 2

## Positive or Negative Number Test:

Positive Numbers > 0
Since 72 ≥ 0 and it is an integer
72 is a positive number

## Whole Number Test:

Positive numbers including 0
with no decimal or fractions
Since 72 ≥ 0 and it is an integer
72 is a whole number

## Prime or Composite Test:

Since 72 has divisors other than 1 and itself
it is a composite number

## Perfect/Deficient/Abundant Test:

Calculate divisor sum D
If D = N, then it's perfect
If D > N, then it's abundant
If D < N, then it's 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 an abundant number!

## Odd or Even Test (Parity Function):

A number is even if it is divisible by 2
If not divisible by 2, it is odd
 36  = 72 2

Since 36 is an integer, 72 is divisible by 2
it is an even number
This can be written as A(72) = Even

## Evil or Odious Test:

Get binary expansion
If binary has even amount 1's, then it's evil
If binary has odd amount 1's, then it's odious
72 to binary = 1001000
There are 2 1's, 72 is an evil number

## Triangular Test:

Can you stack numbers in a pyramid?
Each row above has one item less than the row before it
Using a bottom row of 12 items, we cannot form a pyramid
72 is not triangular

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

## Rectangular Test:

Is there an integer m such that n = m(m + 1)
The integer m = 8 satisifes our rectangular number property.
8(8 + 1) = 72

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

## Automorphic (Curious) Test:

Does n2 ends with n
722 = 72 x 72 = 5184
Since 5184 does not end with 72
it is not automorphic (curious)

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

## Undulating Test:

Do the digits of n alternate in the form abab
Since 72 < 100
We only perform the test on numbers > 99

## Square Test:

Is there a number m such that m2 = n
82 = 64 and 92 = 81 which do not equal 72
Therefore, 72 is not a square

## Cube Test:

Is there a number m such that m3 = n
43 = 64 and 53 = 125 ≠ 72
Therefore, 72 is not a cube

## Palindrome Test:

Is the number read backwards equal to the number?
The number read backwards is 27
Since 72 <> 27
it is not a palindrome

## Palindromic Prime Test:

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

## Repunit Test:

A number is repunit if every digit is equal to 1
Since there is at least one digit in 72 ≠ 1
then it is NOT repunit

## Apocalyptic Power Test:

Does 2n contains the consecutive digits 666.
272 = 4.7223664828696E+21
Since 272 does not have 666
72 is NOT an apocalyptic power

## Pentagonal Test:

It satisfies the form:
 n(3n - 1) 2

## Check values of 7 and 8

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 NOT a pentagonal number

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 NOT a pentagonal number

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

## Hexagonal Test:

Is there 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

## Heptagonal Test:

Is there 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

## Octagonal Test:

Is there 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

## Nonagonal Test:

Is there 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

## Tetrahedral (Pyramidal) Test:

Satisfies the form:
 n(n + 1)(n + 2) 6

## Check values of 6 and 7

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 NOT a tetrahedral (Pyramidal) number

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 NOT a tetrahedral (Pyramidal) number

## Narcissistic (Plus Perfect) Test:

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 digitsm = 72 + 22
Square sum of digitsm = 49 + 4
Square sum of digitsm = 53
Since 53 <> 72
72 is NOT narcissistic (plus perfect)

## Catalan Test:

 Cn  = 2n! (n + 1)!n!

## Check values of 5 and 6

Using n = 6, we have:
 C6  = (2 x 6)! 6!(6 + 1)!

Using our factorial lesson
 C6  = 12! 6!7!

 C6  = 479001600 (720)(5040)

 C6  = 479001600 3628800

C6 = 132
Since this does not equal 72
This is NOT a Catalan number

Using n = 5, we have:
 C5  = (2 x 5)! 5!(5 + 1)!

Using our factorial lesson
 C5  = 10! 5!6!

 C5  = 3628800 (120)(720)

 C5  = 3628800 86400

C5 = 42
Since this does not equal 72
This is NOT a Catalan number

## Property Summary for the number 72

·  Positive
·  Whole
·  Composite
·  Abundant
·  Even
·  Evil
·  Rectangular

Positive
Whole
Composite
Abundant
Even
Evil
Rectangular

### How does the Number Property Calculator work?

This calculator determines if an integer you entered has any of the following properties:
* Even Numbers or Odd Numbers (Parity Function or even-odd numbers)
* Evil Numbers or Odious Numbers
* Perfect Numbers, Abundant Numbers, or Deficient Numbers
* Triangular Numbers
* Prime Numbers or Composite Numbers
* Automorphic (Curious)
* Undulating Numbers
* Square Numbers
* Cube Numbers
* Palindrome Numbers
* Repunit Numbers
* Apocalyptic Power
* Pentagonal
* Tetrahedral (Pyramidal)
* Narcissistic (Plus Perfect)
* Catalan
* Repunit
This calculator has 1 input.

### What 5 formulas are used for the Number Property Calculator?

1. Positive Numbers are greater than 0
2. Whole Numbers are positive numbers, including 0, with no decimal or fractional parts
3. Even numbers are divisible by 2
4. Odd Numbers are not divisible by 2
5. Palindromes have equal numbers when digits are reversed

For more math formulas, check out our Formula Dossier

### What 11 concepts are covered in the Number Property Calculator?

divisor
a number by which another number is to be divided.
even
narcissistic numbers
a given number base b is a number that is the sum of its own digits each raised to the power of the number of digits.
number
an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification. A quantity or amount.
number property
odd
palindrome
A word or phrase which reads the same forwards or backwards
pentagon
a polygon of five angles and five sides
pentagonal number
A number that can be shown as a pentagonal pattern of dots.
n(3n - 1)/2
perfect number
a positive integer that is equal to the sum of its positive divisors, excluding the number itself.
property
an attribute, quality, or characteristic of something