Number 113: Mathematical Properties and Facts
113 is an odd prime number — the 30th prime — with only two factors, 1 and 113. It is a permutable prime (113, 131 and 311 are all prime), a Sophie Germain prime and a Chen prime; it is not a perfect square; and 355/113 ≈ 3.1415929 is the famous fraction for π. In binary it is 1110001, in hex 0x71, in Roman numerals CXIII.
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 |
| Perfect Square? | No (10² = 100, 11² = 121; √113 ≈ 10.63) |
| Strong Number? | No (1! + 1! + 3! = 8) |
| Square of 113 | 12,769 |
| 113 + 113 (double) | 226 |
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 |
For a step-by-step walkthrough of how each of these is built — including base-12, balanced ternary, and base-5 — see 113 in different number systems. The rules behind the Roman form CXIII (and why no other spelling is valid) are covered in 113 in Roman numerals.
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:
- Sophie Germain Prime: 113 is a Sophie Germain prime because 2 × 113 + 1 = 227 is also prime.
- Emirp and Permutable Prime: reversing the digits gives 311, which is prime, and the third arrangement, 131, is prime as well.
- Cousin Prime: 113 forms a cousin prime with 109, as they differ by 4.
- Prime Triplet Member: 113 is the largest member of the prime triplet (107, 109, 113), which follows the pattern p, p+2, p+6.
- 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 Two Squares: 113 = 7² + 8² (49 + 64), which is possible because 113 leaves a remainder of 1 when divided by 4
- Sophie Germain Prime: 2 × 113 + 1 = 227 is also prime
- Chen Prime: 113 + 2 = 115 = 5 × 23 is the product of two primes
- 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 = 7² + 8² (sum of two squares)
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
- Sophie Germain Primes: 113 belongs to this sequence because 2 × 113 + 1 = 227 is prime
- Permutable primes: 113 belongs to the short list of primes whose every digit arrangement is prime (2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 199, 337, …)
- Primes with a prime digit sum: 1 + 1 + 3 = 5
113 in Scientific Contexts
- Chemistry: element 113 is nihonium (Nh), a synthetic element first produced at RIKEN in Japan in 2004 and named in 2016
- Approximating π: 355/113 = 3.14159292…, correct to six decimal places — see 113 as a fraction
Mathematical Puzzles and Challenges with 113
Number 113 Challenge
113 is a prime that leaves remainder 1 when divided by 4, so it can be written as the sum of two perfect squares. Can you find them?
More 113 Mathematical Curiosities
- Repeatedly summing the squares of its digits gives 1² + 1² + 3² = 11, then 2, 4, 16, 37, 58, 89, 145, 42, 20, 4… — the sequence falls into a loop and never reaches 1, so 113 is not a happy number
- 113 can be written as a square plus a prime, for example 2² + 109 (4 + 109) or 4² + 97 (16 + 97)
- The 113th Fibonacci number is 184,551,825,793,033,096,366,333, a number with 24 digits
- 113 is the largest member of the prime triplet (107, 109, 113), which follows the pattern p, p+2, p+6
Fun Facts and Trivia About 113
- 113 seconds is 1 minute and 53 seconds
- $113 in modern U.S. pennies (2.5 g each) would weigh about 28.25 kg (62.3 pounds)
- A stack of 113 U.S. banknotes (about 0.11 mm each) would be roughly 1.2 cm (0.49 inches) tall
- 113 kilometers is approximately 70.2 miles
- 113°F is equal to 45°C, which is considered extremely hot weather
- Element 113 in the periodic table is nihonium (Nh), a synthetic superheavy element confirmed and named in 2016
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 an emirp (311 is also prime); it's a Chen prime; it's a Sophie Germain prime, since 2 × 113 + 1 = 227 is prime; it has a digit sum (5) that is also prime; it can be written as the sum of two squares (7² + 8²); and it's related to element 113 (nihonium) in the periodic table. In number theory, 113 forms a cousin prime pair with 109 and is the largest member of the prime triplet (107, 109, 113).
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.
Is 113 a square number?
No. 113 lies between the perfect squares 100 (10²) and 121 (11²), and its square root, about 10.63, is irrational. It is, however, a sum of two squares: 7² + 8² = 113.
Is 113 a strong number?
No. A strong number equals the sum of the factorials of its digits (like 145 = 1! + 4! + 5!). For 113 the sum is 1! + 1! + 3! = 8.
What is 113 + 113?
113 + 113 = 226, which is 2 × 113. Since 113 is prime, 226 factors as 2 × 113.
What is the next prime number after 113?
The next prime number after 113 is 127, a prime gap of 14. The average gap between primes near 113 is only about ln(113) ≈ 4.7, and 113–127 is the first place on the number line where a gap of 14 (or more) occurs, so it is a record-setting "maximal" prime gap.
How does 113 relate to other mathematical constants?
355/113 ≈ 3.1415929 is 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...).