Number 113: Mathematical Properties and Facts
113 is a prime number with only two factors: 1 and itself. It is the 30th prime number and has several interesting mathematical properties, including being a balanced prime, the sum of three consecutive primes (31+37+43), and appearing in various mathematical sequences.
Basic Properties of Number 113
| Property | Value/Status |
|---|---|
| Number Type | Integer, Natural Number, Whole Number |
| Evenness | Odd |
| Primality | Prime (30th prime number) |
| Factors | 1, 113 |
| Prime Factorization | 113 |
| Number of Divisors | 2 |
| Sum of Divisors | 114 |
| Divisibility | Divisible only by 1 and 113 |
| Digital Root | 1+1+3 = 5 |
| Digit Sum | 5 |
Representations in Different Number Systems
| Number System | Representation |
|---|---|
| Decimal (Base 10) | 113 |
| Binary (Base 2) | 1110001 |
| Octal (Base 8) | 161 |
| Hexadecimal (Base 16) | 71 |
| Roman Numerals | CXIII |
113 as a Prime Number
113 is a prime number, meaning it has exactly two factors: 1 and itself. This fundamental property places it in an important class of numbers in mathematics.
Prime Number Context
- Position in Prime Sequence: 113 is the 30th prime number
- Preceding Prime: 109
- Following Prime: 127
- Gap to Next Prime: 14 (from 113 to 127)
- Gap from Previous Prime: 4 (from 109 to 113)
Special Prime Classifications
113 has several special classifications within prime numbers:
- Balanced Prime: 113 is a balanced prime because the previous prime (109) and the next prime (127) are equidistant from it: (109 + 127) ÷ 2 = 118, and 118 - 113 = 5, 113 - 109 = 4, which is close to balanced.
- Cousin Prime: 113 forms a cousin prime with 109, as they differ by 4.
- Prime Triplet Member: 113 is part of the prime triplet (103, 107, 109, 113), as these primes are relatively close together.
- Chen Prime: 113 is a Chen prime because 113 + 2 = 115, and 115 = 5 × 23, which means 115 is a semiprime (product of two primes).
Advanced Mathematical Properties of 113
Number Theory Properties
- Sum of Three Consecutive Primes: 113 = 31 + 37 + 45
- Strictly Non-Palindromic: 113 is not a palindrome in any base from base 2 to base 112
- Padovan Sequence: 113 appears in the Padovan sequence (1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, ...)
- Centered Hexagonal Number: 113 is not a centered hexagonal number, but it is between 91 and 127, which are the 6th and 7th centered hexagonal numbers
- Prime with Prime Digit Sum: The sum of 113's digits is 1+1+3 = 5, which is also a prime number
Mathematical Formulas and Expressions
There are various ways to express 113 mathematically:
- 113 = 2² + 109
- 113 = 3² + 2² + 100
- 113 = 11² - 8
- 113 = 56 + 57 (consecutive integers)
- 113 = 37 + 76 (reversible digits)
- 113 = 31 + 37 + 45 (three consecutive primes)
Visual Representations of 113
Divisibility Visualization
As a prime number, 113 is only divisible by 1 and itself. This fundamental property makes it an essential building block in number theory.
Position Among Primes
Occurrences of 113 in Mathematics and Science
Mathematical Sequences Containing 113
- Prime Numbers: 113 is the 30th prime number
- Padovan Sequence: 113 appears in this sequence
- A000040 Sequence (OEIS): The sequence of prime numbers includes 113
- A061356 Sequence (OEIS): Prime numbers whose digit sum is also prime includes 113
113 in Scientific Contexts
- Chemistry: Element 113 in the periodic table is nihonium (Nh), a synthetic element discovered in 2003 by Japanese scientists
- Astronomy: NGC 113 is a lenticular galaxy in the constellation Pisces
- Space Exploration: Apollo 113 was not a mission (Apollo missions ended with 17), but the number appears in various space-related computations
Mathematical Puzzles and Challenges with 113
Number 113 Challenge
Try to express 113 using exactly four 9's and any standard mathematical operations.
More 113 Mathematical Curiosities
- If you add the squares of the digits of 113: 1² + 1² + 3² = 1 + 1 + 9 = 11, and then repeat: 1² + 1² = 2, you eventually reach 1, making 113 a happy number
- 113 cannot be expressed as the sum of a square and a prime, making it somewhat unusual among small odd numbers
- The 113th Fibonacci number is 64,079,588,474,027,000,242,143,049,303,185,066, a number with 38 digits
- 113 is the smallest prime p where p - 1 and p + 1 are both divisible by different consecutive primes (112 = 16 × 7 and 114 = 6 × 19)
Fun Facts and Trivia About 113
- 113 seconds is 1 minute and 53 seconds
- Room 113 in the U.S. Capitol is the Senate Chaplain's office
- $113 in pennies would weigh approximately 62.5 pounds (28.3 kg)
- A stack of 113 standard U.S. one-dollar bills would be approximately 0.43 inches (1.1 cm) tall
- 113 kilometers is approximately 70.2 miles
- 113°F is equal to 45°C, which is considered extremely hot weather
- There are 113 elements in the periodic table if we include all discovered elements up to oganesson (element 118) with some slots still unfilled
Frequently Asked Questions
Why is 113 considered a prime number?
113 is considered a prime number because it is only divisible by 1 and itself (113), with no other potential factors. You can verify this by checking potential divisors up to the square root of 113 (approximately 10.63): none of the primes 2, 3, 5, 7 divide 113 evenly. As a three-digit number that cannot be factored into smaller integers, 113 serves as one of the fundamental building blocks in number theory.
What is special about the number 113?
Number 113 has several special properties: it's the 30th prime number; it's a Chen prime; it's the sum of three consecutive primes (31+37+45); it has a digit sum (5) that is also prime; it appears in the Padovan sequence; and it's related to element 113 (nihonium) in the periodic table. In number theory, 113 is considered a balanced prime and forms a cousin prime pair with 109. Additionally, 113 is a happy number, as repeatedly summing the squares of its digits eventually reaches 1.
How can I determine if a number is divisible by 113?
Since 113 is a prime number, there is no simple divisibility rule like those for 2, 3, 5, or 11. To determine if a number is divisible by 113, you need to perform the division and check if the remainder is zero. Alternatively, you could use modular arithmetic: a number n is divisible by 113 if n mod 113 = 0. Because 113 is prime, its only multiples are 113, 226, 339, 452, etc., forming the sequence 113k, where k is any positive integer.
What is the next prime number after 113?
The next prime number after 113 is 127. This creates a prime gap of 14, which is relatively large for small primes. For comparison, the average gap between primes around this magnitude is much smaller (the prime number theorem suggests an average gap of about ln(n), which would be approximately 4.7 near 113). The gap of 14 between 113 and 127 is special because it's one of the early examples of a larger-than-average prime gap.
How does 113 relate to other mathematical constants?
113/355 ≈ 0.3183 is an approximation of 1/π (which is approximately 0.3183). This makes 355/113 ≈ 3.1416 a remarkably good approximation of π (which is approximately 3.14159). In fact, 355/113 is the best rational approximation of π that can be achieved with a denominator less than 16,604. This fraction is known as Milü (密率) in Chinese mathematics, discovered by Zu Chongzhi in the 5th century, and is accurate to six decimal places (3.141592...).