It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 397, the answer is: yes, 397 is a prime number because it has only two distinct divisors: 1 and itself (397).
As a consequence, 397 is only a multiple of 1 and 397..
Since 397 is a prime number, 397 is also a deficient number, that is to say 397 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 397 without 397 itself (that is 1, by definition!).
Parity of 397
397 is an odd number, because it is not evenly divisible by 2.
Is 397 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 397 is about 19.925.
Thus, the square root of 397 is not an integer, and therefore 397 is not a square number.
Anyway, 397 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 397?
The square of a number (here 397) is the result of the product of this number (397) by itself (i.e., 397 × 397); the square of 397 is sometimes called "raising 397 to the power 2", or "397 squared".
As a consequence, 397 is the square root of 157 609.
Number of digits of 397
397 is a number with 3 digits.
What are the multiples of 397?
The multiples of 397 are all integers evenly divisible by 397, that is all numbers such that the remainder of the division by 397 is zero. There are infinitely many multiples of 397. The smallest multiples of 397 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 397 too, since 0 × 397 = 0
- 397: indeed, 397 is a multiple of itself, since 397 is evenly divisible by 397 (we have 397 / 397 = 1, so the remainder of this division is indeed zero)
- 794: indeed, 794 = 397 × 2
- 1 191: indeed, 1 191 = 397 × 3
- 1 588: indeed, 1 588 = 397 × 4
- 1 985: indeed, 1 985 = 397 × 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 397). 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 19.925). 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.