# Numerical properties of 45

## Enter Integer

Show numerical properties of 45

We start by listing out divisors for 45
DivisorDivisor Math
145 ÷ 1 = 45
345 ÷ 3 = 15
545 ÷ 5 = 9
945 ÷ 9 = 5
1545 ÷ 15 = 3

## Positive or Negative Number Test:

Positive Numbers > 0
Since 45 ≥ 0 and it is an integer
45 is a positive number

## Whole Number Test:

Positive numbers including 0
with no decimal or fractions
Since 45 ≥ 0 and it is an integer
45 is a whole number

## Prime or Composite Test:

Since 45 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 + 3 + 5 + 9 + 15
Divisor Sum = 33
Since our divisor sum of 33 < 45
45 is a deficient 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
 22.5  = 45 2

Since 22.5 is not an integer, 45 is not divisible by
it is an odd number
This can be written as A(45) = Odd

## 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
45 to binary = 101101
There are 4 1's, 45 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 9 items, 45 forms a triangle
It is a triangular number

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)
No integer m exists such that m(m + 1) = 45
45 is not rectangular

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

## Automorphic (Curious) Test:

Does n2 ends with n
452 = 45 x 45 = 2025
Since 2025 does not end with 45
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 45 < 100
We only perform the test on numbers > 99

## Square Test:

Is there a number m such that m2 = n
62 = 36 and 72 = 49 which do not equal 45
Therefore, 45 is not a square

## Cube Test:

Is there a number m such that m3 = n
33 = 27 and 43 = 64 ≠ 45
Therefore, 45 is not a cube

## Palindrome Test:

Is the number read backwards equal to the number?
The number read backwards is 54
Since 45 <> 54
it is not a palindrome

## Palindromic Prime Test:

Is it both prime and a palindrome
From above, since 45 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 45 ≠ 1
then it is NOT repunit

## Apocalyptic Power Test:

Does 2n contains the consecutive digits 666.
245 = 35184372088832
Since 245 does not have 666
45 is NOT an apocalyptic power

## Pentagonal Test:

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

## Check values of 5 and 6

Using n = 6, we have:
 6(3(6 - 1) 2

 6(18 - 1) 2

 6(17) 2

 102 2

51 ← Since this does not equal 45
this is NOT a pentagonal number

Using n = 5, we have:
 5(3(5 - 1) 2

 5(15 - 1) 2

 5(14) 2

 70 2

35 ← Since this does not equal 45
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)
The integer m = 5 is hexagonal
Since 5(2(5) - 1) = 45

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 = 45
Therefore 45 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) = 45
Therefore 45 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 = 45
Therefore 45 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 5 and 6

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

 6(7)(8) 6

 336 6

56 ← Since this does not equal 45
This is NOT a tetrahedral (Pyramidal) number

Using n = 5, we have:
 5(5 + 1)(5 + 2) 6

 5(6)(7) 6

 210 6

35 ← Since this does not equal 45
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
45 is a 2 digit number, so m = 2
Square sum of digitsm = 42 + 52
Square sum of digitsm = 16 + 25
Square sum of digitsm = 41
Since 41 <> 45
45 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 45
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 45
This is NOT a Catalan number

## Property Summary for the number 45

·  Positive
·  Whole
·  Composite
·  Deficient
·  Odd
·  Evil
·  Triangular
·  Hexagonal

Positive
Whole
Composite
Deficient
Odd
Evil
Triangular
Hexagonal

### 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