Numerical properties of 4

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Show numerical properties of 4

We start by listing out divisors for 4

Divisor	Divisor Math
1	4 ÷ 1 = 4
2	4 ÷ 2 = 2

Positive or Negative Number Test:

Positive Numbers > 0

Since 4 ≥ 0 and it is an integer
4 is a positive number

Whole Number Test:

Positive numbers including 0
with no decimal or fractions

Since 4 ≥ 0 and it is an integer
4 is a whole number

Prime or Composite Test:

Since 4 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
Divisor Sum = 3
Since our divisor sum of 3 < 4
4 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

2  =	4
	2

Since 2 is an integer, 4 is divisible by 2
it is an even number
This can be written as A(4) = 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
4 to binary = 100
There are 1 1's, 4 is an odious 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 3 items, we cannot form a pyramid
4 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)
No integer m exists such that m(m + 1) = 4
4 is not rectangular

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

Automorphic (Curious) Test:

Does n² ends with n
4² = 4 x 4 = 16
Since 16 does not end with 4
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 4 < 100
We only perform the test on numbers > 99

Square Test:

Is there a number m such that m² = n
2² = 4
Since 4 is the square of 2
4 is a square

Cube Test:

Is there a number m such that m³ = n
1³ = 1 and 2³ = 8 ≠ 4
Therefore, 4 is not a cube

Palindrome Test:

Is the number read backwards equal to the number?
The number read backwards is 4
Since 4 is the same backwards and forwards
it is a palindrome

Palindromic Prime Test:

Is it both prime and a palindrome
From above, since 4 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 4 ≠ 1
then it is NOT repunit

Apocalyptic Power Test:

Does 2ⁿ contains the consecutive digits 666.
2⁴ = 16
Since 2⁴ does not have 666
4 is NOT an apocalyptic power

Pentagonal Test:

It satisfies the form:

n(3n - 1)
2

Check values of 1 and 2

Using n = 2, we have:

2(3(2 - 1)
2

2(6 - 1)
2

2(5)
2

10
2

5 ← Since this does not equal 4
this is NOT a pentagonal number

Using n = 1, we have:

1(3(1 - 1)
2

1(3 - 1)
2

1(2)
2

2
2

1 ← Since this does not equal 4
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) = 4
Therefore 4 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 = 4
Therefore 4 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) = 4
Therefore 4 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 = 4
Therefore 4 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

Using n = 2, we have:

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

2(3)(4)
6

24
6

4 ← Since this equals 4
This is a tetrahedral (Pyramidal)number

Narcissistic (Plus Perfect) Test:

Is equal to the square sum of it's m-th powers of its digits
4 is a 1 digit number, so m = 1
Square sum of digits^m = 4¹
Square sum of digits^m = 4
Square sum of digits^m = 4
Since 4 = 4
4 is narcissistic (plus perfect)

Catalan Test:

Cn  =	2n!
	(n + 1)!n!

Check values of 2 and 3

Using n = 3, we have:

C3  =	(2 x 3)!
	3!(3 + 1)!

Using our factorial lesson

C3  =	6!
	3!4!

C3  =	720
	(6)(24)

C3  =	720
	144

C₃ = 5
Since this does not equal 4
This is NOT a Catalan number

Using n = 2, we have:

C2  =	(2 x 2)!
	2!(2 + 1)!

Using our factorial lesson

C2  =	4!
	2!3!

C2  =	24
	(2)(6)

C2  =	24
	12

C₂ = 2
Since this does not equal 4
This is NOT a Catalan number

Property Summary for the number 4

  ·  Positive
  ·  Whole
  ·  Composite
  ·  Deficient
  ·  Even
  ·  Odious
  ·  Square
  ·  Palindrome
  ·  Tetrahedral (Pyramidal)
  ·  Narcissistic (Plus Perfect)

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Tags:

• divisor
• even
• narcissistic numbers
• number property
• number
• odd
• palindrome
• pentagon
• pentagonal number
• perfect number
• property

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