# Numerical properties of 30

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Determine the numerical properties of 30

We start by listing out divisors for 30
DivisorDivisor Math
130 ÷ 1 = 30
230 ÷ 2 = 15
330 ÷ 3 = 10
530 ÷ 5 = 6
630 ÷ 6 = 5
1030 ÷ 10 = 3
1530 ÷ 15 = 2

Prime or Composite Test:
Since 30 has divisors other than 1 and itself, it is a composite number

Perfect/Deficient/Abundant Test:
A perfect number is a number who has a divisor sum equal to itself. An abundant number is a number who has a divisor sum greater than the number, otherwise, it is 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, else it is odd
 15 = 30 2

Since 15 is an integer, 30 is divisible by 2, and therefore, it is an even number
This can be written as A(30) = Even

Evil or Odious Test:
A number is evil is there are an even number of 1's in the binary expansion, else, it is odious.
Using our decimal to binary calculator, we see the binary expansion of 30 is 11110
Since there are 4 1's in the binary expansion which is an even number, 30 is an evil number

Triangular Test:
A number is triangular if it can be stacked in a pyramid with each row above containing one item less than the row before it, ending with 1 item at the top
Using a bottom row of 8 items, we cannot form a pyramid with our numbers, therefore 30 is not triangular

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

Rectangular Test:
A number n is rectangular is if there is an integer m such that n = m(m + 1)
The integer m = 5 satisifes our rectangular number property, since 5(5 + 1) = 30

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

Automorphic (Curious) Test:
A number (n) is automorphic (curious) if n2 ends with n
302 = 30 x 30 = 900
Since 900 does not end with 30, it is not automorphic (curious)

Undulating Test:
A number (n) is undulating if the digits that comprise n alternate in the form abab
Since 30 < 100, we only perform the test on 3 digit numbers or higher

Square Test:
A number (n) is a square if there exists a number m such that m2 = n
Analyzing squares, we see that 52 = 25 and 62 = 36 which do not equal 30
Therefore, 30 is not a square

Cube Test:
A number (n) is a cube if there exists a number m such that m3 = n
Analyzing cubes, we see that 33 = 27 and 43 = 64 which do not equal 30
Therefore, 30 is not a cube

Palindrome Test:
A number (n) is a palindrome if the number read backwards equals the number itself
The number read backwards is 03
Since 30 <> 03, it is not a palindrome

Palindromic Prime Test:
A number is a palindromic prime if it is both prime and a palindrome
From above, since 30 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 30 not equal to 1, then it is NOT repunit

Apocalyptic Power Test:
A number (n) is apocalyptic power if 2n contains the consecutive digits 666.
230 = 1073741824
Since 230 does not contain the sequence 666, 30 is NOT an apocalyptic power

Pentagonal Test:
A pentagonal number is one which satisfies the form:
 n(3n - 1) 2

Check values of 4 and 5
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 30, this is NOT a pentagonal number

Using n = 4, we have:
 4(3(4 - 1) 2

 4(12 - 1) 2

 4(11) 2

 44 2

22 ← Since this does not equal 30, this is NOT a pentagonal number

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

Hexagonal Test:
A number n is hexagonal is if there is an integer m such that n = m(2m - 1)
No integer m exists such that m(2m - 1) = 30, therefore 30 is not hexagonal

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

Heptagonal Test:
A number n is heptagonal is if there is an integer m such that:
 m = n(5n - 3) 2

No integer m exists such that m(5m - 3)/2 = 30, therefore 30 is not heptagonal

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

Octagonal Test:
A number n is octagonal is if there is an integer m such that n = m(3m - 3)
No integer m exists such that m(3m - 2) = 30, therefore 30 is not octagonal

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

Nonagonal Test:
A number n is nonagonal is if there is an integer m such that:
 m = n(7n - 5) 2

No integer m exists such that m(7m - 5)/2 = 30, therefore 30 is not nonagonal

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

Tetrahedral (Pyramidal) Test:
A tetrahedral (Pyramidal) number is one which satisfies the form:
 n(n + 1)(n + 2) 6

Check values of 4 and 5
Using n = 5, we have:
 5(5 + 1)(5 + 2) 6

 5(6)(7) 6

 210 6

35 ← Since this does not equal 30, this is NOT a tetrahedral (Pyramidal) number

Using n = 4, we have:
 4(4 + 1)(4 + 2) 6

 4(5)(6) 6

 120 6

20 ← Since this does not equal 30, this is NOT a tetrahedral (Pyramidal) number

Narcissistic (Plus Perfect) Test:
An m digit number n is narcissistic if it is equal to the square sum of it's m-th powers of its digits
30 is a 2 digit number, so m = 2
Square sum of digitsm = 32 + 02
Square sum of digitsm = 9 + 0
Square sum of digitsm = 9
Since 9 <> 30, 30 is NOT narcissistic (plus perfect)

Catalan Test:
The nth Catalan number Cn is denoted by:
 Cn = 2n! (n + 1)!n!

Check values of 4 and 5
Using n = 5, we have:
 C5 = (2 x 5)! 5!(5 + 1)!

Using our factorial lesson to evaluate, we get
 C5 = 10! 5!6!

 C5 = 3628800 (120)(720)

 C5 = 3628800 86400

C5 = 42
Since this does not equal 30, this is NOT a Catalan number

Using n = 4, we have:
 C4 = (2 x 4)! 4!(4 + 1)!

Using our factorial lesson to evaluate, we get
 C4 = 8! 4!5!

 C4 = 40320 (24)(120)

 C4 = 40320 2880

C4 = 14
Since this does not equal 30, this is NOT a Catalan number

Property Summary for the number 30
· Composite
· Abundant
· Even
· Evil
· Rectangular