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:
Rectangular Test:
Is there an integer m such that n = m(m + 1)
No integer m exists such that m(m + 1) = 10thtriangularnumber
10thtriangularnumber is not rectangular
Rectangular number:
Automorphic (Curious) Test:
Does n2 ends with n
10threctangularnumber2 = 10threctangularnumber x 10threctangularnumber = 100
Since 100 does not end with 10threctangularnumber
it is not automorphic (curious)
Automorphic number:
Undulating Test:
Do the digits of n alternate in the form abab
Since 10thautomorphicnumber < 100
We only perform the test on numbers > 99
Square Test:
Is there a number m such that m2 = n?
32 = 9 and 42 = 16 which do not equal 10thautomorphicnumber
Therefore, 10thautomorphicnumber is not a square
Cube Test:
Is there a number m such that m3 = n
23 = 8 and 33 = 27 ≠ 10thautomorphicnumber
Therefore, 10thautomorphicnumber is not a cube
Palindrome Test:
Is the number read backwards equal to the number?
The number read backwards is rebmuncihpromotuaht01
Since 10thautomorphicnumber <> rebmuncihpromotuaht01
it is not a palindrome
Palindromic Prime Test:
Is it both prime and a palindrome
From above, since 10thautomorphicnumber 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 10thautomorphicnumber ≠ 1
then it is NOT repunit
Apocalyptic Power Test:
Does 2n contain the consecutive digits 666?
210thautomorphicnumber = 1024
Since 210thautomorphicnumber does not have 666
10thautomorphicnumber is NOT an apocalyptic power
Pentagonal Test:
It satisfies the form:
| n(3n - 1) | |
| 2 |
Check values of 2 and 3
Using n = 3, we have:
| 3(3(3 - 1) | |
| 2 |
| 3(9 - 1) | |
| 2 |
| 3(8) | |
| 2 |
| 24 | |
| 2 |
12 ← Since this does not equal 10thautomorphicnumber
this is NOT a pentagonal number
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 10thautomorphicnumber
this is NOT a pentagonal number
Pentagonal number:
Hexagonal Test:
Is there an integer m such that n = m(2m - 1)
No integer m exists such that m(2m - 1) = 10thpentagonalnumber
Therefore 10thpentagonalnumber is not hexagonal
Hexagonal number:
Heptagonal Test:
Is there an integer m such that:
| m = | n(5n - 3) |
| 2 |
No integer m exists such that m(5m - 3)/2 = 10thhexagonalnumber
Therefore 10thhexagonalnumber is not heptagonal
Heptagonal number:
Octagonal Test:
Is there an integer m such that n = m(3m - 3)
No integer m exists such that m(3m - 2) = 10thheptagonalnumber
Therefore 10thheptagonalnumber is not octagonal
Octagonal number:
Nonagonal Test:
Is there an integer m such that:
| m = | n(7n - 5) |
| 2 |
No integer m exists such that m(7m - 5)/2 = 10thoctagonalnumber
Therefore 10thoctagonalnumber is not nonagonal
Nonagonal number:
Tetrahedral (Pyramidal) Test:
Tetrahederal numbers satisfy the form:
| n(n + 1)(n + 2) | |
| 6 |
Using n = 3, we have:
| 3(3 + 1)(3 + 2) | |
| 6 |
| 3(4)(5) | |
| 6 |
| 60 | |
| 6 |
10 ← Since this equals 10thnonagonalnumber
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
10thnonagonalnumber is a 19 digit number, so m = 19
Square sum of digitsm = 119 + 019 + t19 + h19 + n19 + o19 + n19 + a19 + g19 + o19 + n19 + a19 + l19 + n19 + u19 + m19 + b19 + e19 + r19
Square sum of digitsm = 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0
Square sum of digitsm = 1
Since 1 <> 10thnonagonalnumber
10thnonagonalnumber is NOT narcissistic (plus perfect)
Catalan Test:
| Cn = | 2n! |
| (n + 1)!n! |
Check values of 3 and 4
Using n = 4, we have:
| C4 = | (2 x 4)! |
| 4!(4 + 1)! |
Using our factorial lesson
| C4 = | 8! |
| 4!5! |
| C4 = | 40320 |
| (24)(120) |
| C4 = | 40320 |
| 2880 |
C4 = 14
Since this does not equal 10thnonagonalnumber
This is NOT a Catalan number
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 10thnonagonalnumber
This is NOT a Catalan number
Number Properties for 10thnonagonalnumber
Final Answer
Whole
Composite
Deficient
Even
Odious
Tetrahedral (Pyramidal)
