It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 313, the answer is: yes, 313 is a prime number because it has only two distinct divisors: 1 and itself (313).
As a consequence, 313 is only a multiple of 1 and 313..
Since 313 is a prime number, 313 is also a deficient number, that is to say 313 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 313 without 313 itself (that is 1, by definition!).
Parity of 313
313 is an odd number, because it is not evenly divisible by 2.
Is 313 a perfect square number?
A number is a perfect square (or a square number) if its square root is an integer; that is to say, it is the product of an integer with itself. Here, the square root of 313 is about 17.692.
Thus, the square root of 313 is not an integer, and therefore 313 is not a square number.
Anyway, 313 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 313?
The square of a number (here 313) is the result of the product of this number (313) by itself (i.e., 313 × 313); the square of 313 is sometimes called "raising 313 to the power 2", or "313 squared".
As a consequence, 313 is the square root of 97 969.
Number of digits of 313
313 is a number with 3 digits.
What are the multiples of 313?
The multiples of 313 are all integers evenly divisible by 313, that is all numbers such that the remainder of the division by 313 is zero. There are infinitely many multiples of 313. The smallest multiples of 313 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 313 too, since 0 × 313 = 0
- 313: indeed, 313 is a multiple of itself, since 313 is evenly divisible by 313 (we have 313 / 313 = 1, so the remainder of this division is indeed zero)
- 626: indeed, 626 = 313 × 2
- 939: indeed, 939 = 313 × 3
- 1 252: indeed, 1 252 = 313 × 4
- 1 565: indeed, 1 565 = 313 × 5
How to determine whether an integer is a prime number?
To determine the primality of a number, several algorithms can be used. The most naive technique is to test all divisors strictly smaller to the number of which we want to determine the primality (here 313). First, we can eliminate all even numbers greater than 2 (and hence 4, 6, 8…). Then, we can stop this check when we reach the square root of the number of which we want to determine the primality (here the square root is about 17.692). Historically, the sieve of Eratosthenes (dating from the Greek mathematics) implements this technique in a relatively efficient manner.
More modern techniques include the sieve of Atkin, probabilistic algorithms, and the cyclotomic AKS test.