Show numerical properties of 30
We start by listing out divisors for 30
| Divisor | Divisor Math |
|---|---|
| 1 | 30 ÷ 1 = 30 |
| 2 | 30 ÷ 2 = 15 |
| 3 | 30 ÷ 3 = 10 |
| 5 | 30 ÷ 5 = 6 |
| 6 | 30 ÷ 6 = 5 |
| 10 | 30 ÷ 10 = 3 |
| 15 | 30 ÷ 15 = 2 |
Positive or Negative Number Test:
Positive Numbers > 0
Since 30 ≥ 0 and it is an integer
30 is a positive number
Whole Number Test:
Positive numbers including 0
with no decimal or fractions
Since 30 ≥ 0 and it is an integer
30 is a whole number
Prime or Composite Test:
Since 30 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 + 5 + 6 + 10 + 15
Divisor Sum = 42
Since our divisor sum of 42 > 30
30 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
| 15 = | 30 |
| 2 |
Since 15 is an integer, 30 is divisible by 2
it is an even number
This can be written as A(30) = 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
30 to binary = 11110
There are 4 1's, 30 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 8 items, we cannot form a pyramid
30 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
Abundant
Even
Evil
Tetrahedral (Pyramidal)
Common Core State Standards In This Lesson
What is the Answer?
Whole
Composite
Abundant
Even
Evil
Tetrahedral (Pyramidal)
How does the Number Property Calculator work?
* 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?
Whole Numbers are positive numbers, including 0, with no decimal or fractional parts
Even numbers are divisible by 2
Odd Numbers are not divisible by 2
Palindromes have equal numbers when digits are reversed
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
