# 36

Jump to navigation
Jump to search

## Contents

## Number

$36$ (**thirty-six**) is:

- $2^2 \times 3^2$

- The $2$nd power of $6$ after $(1)$, $6$:
- $36 = 6^2$

- The $2$nd number after $1$ to be both square and triangular:
- $36 = 6^2 = \dfrac {8 \times \paren {8 + 1} } 2$

- The $3$rd of three $2$-digit integers divisible by both the sum and product of its digits:
- $36 = \paren {3 + 6} \times 4 = \paren {3 \times 6} \times 2$

- The $6$th square number after $1$, $4$, $9$, $16$, $25$:
- $36 = 6 \times 6$

- The $6$th abundant number after $12$, $18$, $20$, $24$, $30$:
- $1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55 > 36$

- The $7$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$ such that no factorial of an integer can end with $n$ zeroes.

- The $7$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$:
- $\dfrac {\map \sigma {36} } {36} = \dfrac {91} {36} = 2 \cdotp 52 \dot 7$

- The $7$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$:
- $\map \tau {36} = 9$

- The $8$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$:
- $36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = \dfrac {8 \times \paren {8 + 1} } 2$

- The $8$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$:
- $36 = 3 + 6 + 9 + 18$

- The $9$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$

- The $13$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$:
- $36 = 8 + 28$

- The $14$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$:
- $\map \sigma {36} = 91$

- The $14$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$:
- $36 = 2 \times 18 = 2 \times \left({3 \times 6}\right)$

- The $14$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
- $3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $\ldots$

- The $18$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$, $20$, $21$, $24$, $27$, $30$:
- $36 = 4 \times 9 = 4 \times \paren {3 + 6}$

- The $20$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$, $32$, $35$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

- The $26$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
- $2^{36} = 68 \, 719 \, 476 \, 736$

- The total of all the entries in a magic cube of order $2$ (if it were to exist), after $1$:
- $36 = \displaystyle \sum_{k \mathop = 1}^{2^3} k = \dfrac {2^3 \paren {2^3 + 1} } 2$

- $35$ and $4374$ have the same prime factors between them as $36$ and $4375$:
- $35 = 5 \times 7$, $4374 = 2 \times 3^7$; $36 = 2^2 \times 3^2$, $4375 = 5^4 \times 7$

## Also see

*Previous ... Next*: Highly Composite Number*Previous ... Next*: Superabundant Number*Previous ... Next*: Zuckerman Number*Previous*: 2-Digit Numbers divisible by both Product and Sum of Digits

*Previous ... Next*: Ulam Number*Previous ... Next*: Triangular Number

*Previous ... Next*: 91 is Pseudoprime to 35 Bases less than 91*Previous ... Next*: Abundant Number*Previous ... Next*: Semiperfect Number*Previous ... Next*: Harshad Number*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Numbers of Zeroes that Factorial does not end with

*Previous ... Next*: Powerful Number

*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation

## Historical Note

The number $36$ was held in particularly high regard by the ancient Greeks, as it is the sum of the first $4$ even numbers and the first $4$ odd numbers.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $36$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $36$

Categories:

- Powers of 6/Examples
- Highly Composite Numbers/Examples
- Superabundant Numbers/Examples
- Zuckerman Numbers/Examples
- Square Numbers/Examples
- Ulam Numbers/Examples
- Triangular Numbers/Examples
- Abundant Numbers/Examples
- Semiperfect Numbers/Examples
- Harshad Numbers/Examples
- Highly Abundant Numbers/Examples
- Powerful Numbers/Examples
- Specific Numbers
- 36