Is 113 a Prime Number?
Yes, 113 is a prime number. It is only divisible by 1 and itself (113), which is the definition of a prime number.
Why 113 Is Prime
To determine if 113 is prime, we need to check if it has any divisors other than 1 and itself. A number is prime if and only if it has exactly two divisors: 1 and itself.
Divisibility Check for 113
Let's check if 113 is divisible by any number from 2 to the square root of 113 (which is approximately 10.63):
When we test potential divisors from 2 up to the square root of 113 (approximately 10.63), we find that none of them divide 113 evenly:
- 113 ÷ 2 = 56.5 (remainder 1, not divisible)
- 113 ÷ 3 = 37.67... (not divisible)
- 113 ÷ 5 = 22.6 (not divisible)
- 113 ÷ 7 = 16.14... (not divisible)
- 113 ÷ 11 = 10.27... (not divisible)
Since no number between 2 and the square root of 113 divides it evenly, 113 is indeed a prime number.
Mathematical Properties of 113
Beyond being a prime number, 113 has several interesting mathematical properties:
- 113 is the 30th prime number (following 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113)
- It is an emirp (a prime number that results in a different prime when its digits are reversed) because 311 is also prime
- 113 is a balanced prime because it is the arithmetic mean of the two neighboring primes (109 and 117); however, 117 is not prime, so this doesn't apply
- The sum of its digits (1+1+3 = 5) is a prime number
113 in Number Theory
In number theory, prime numbers like 113 have special significance:
- Prime numbers are the "building blocks" of all integers through the Fundamental Theorem of Arithmetic
- 113 cannot be expressed as the sum of two squares (unlike some primes like 5 = 1² + 2²)
- 113 is a prime factor in numerous mathematical expressions and equations
- It is one of the primes in the sequence of "isolated primes" (primes that are not part of a twin prime pair)
Frequently Asked Questions
Is 113 a Composite Number?
No, 113 is not a composite number. It is a prime number because it has exactly two factors: 1 and itself (113).
What Are the Factors of 113?
The factors of 113 are 1 and 113. As a prime number, 113 has exactly two factors.
Is 113 a Perfect Number?
No, 113 is not a perfect number. A perfect number equals the sum of its proper divisors, and the only proper divisor of 113 is 1, which is far less than 113.
What is Special About the Number 113?
113 is special because it's a prime number, the 30th prime number specifically. It's also an emirp (remains prime when its digits are reversed to form 311).