Is 113 a Prime Number?
Yes, 113 is a prime number. Its only divisors are 1 and 113, so it is prime, not composite. It is the 30th prime number, and it is not divisible by 2, 3, 5 or 7 — the only primes you need to test, because √113 ≈ 10.63.
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):
You only need to test prime divisors up to the square root of 113 (approximately 10.63): if 113 had a factor larger than 10.63, it would also need a matching factor smaller than 10.63. The primes to check are therefore 2, 3, 5 and 7:
- 113 ÷ 2 = 56 remainder 1 — 113 is odd, so not divisible
- 113 ÷ 3 = 37 remainder 2 — digit sum 1+1+3 = 5 is not a multiple of 3
- 113 ÷ 5 = 22 remainder 3 — it does not end in 0 or 5
- 113 ÷ 7 = 16 remainder 1 — 7 × 16 = 112, one short of 113
Since none of these primes divides 113 evenly, no other number (apart from 1 and 113) can either, so 113 is a prime number. Testing 11 is unnecessary: 11 × 11 = 121 is already larger than 113.
For the full divisor list see factors of 113; for a broader overview see properties of the number 113. A Spanish version of this page is available: ¿Es 113 un número primo?
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
- It is a permutable prime: every rearrangement of its digits — 113, 131 and 311 — is prime
- It is a Sophie Germain prime because 2 × 113 + 1 = 227 is also prime
- Its neighboring primes are 109 and 127, so 113 sits at the start of a prime gap of 14
- 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 can be expressed as the sum of two squares: 113 = 7² + 8² (49 + 64), which is possible for every prime that leaves a remainder of 1 when divided by 4
- The fraction 355/113 ≈ 3.1415929 approximates π to six decimal places; it is known as Milü, after the 5th-century Chinese mathematician Zu Chongzhi
- 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.
What Is 113 Divisible By?
113 is divisible only by 1 and 113. Dividing it by any other whole number from 2 to 112 leaves a remainder.
Is 113 a Square Number?
No. 10² = 100 and 11² = 121, so 113 lies between two consecutive perfect squares. Its square root is about 10.63, which is irrational. See square root of 113.
Is 113 a Strong Number?
No. A strong number equals the sum of the factorials of its digits (for example 145 = 1! + 4! + 5!). For 113 that sum is 1! + 1! + 3! = 1 + 1 + 6 = 8, not 113.
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).