It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 197, the answer is: yes, 197 is a prime number because it has only two distinct divisors: 1 and itself (197).
As a consequence, 197 is only a multiple of 1 and 197..
Since 197 is a prime number, 197 is also a deficient number, that is to say 197 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 197 without 197 itself (that is 1, by definition!).
Parity of 197
197 is an odd number, because it is not evenly divisible by 2.
Is 197 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 197 is about 14.036.
Thus, the square root of 197 is not an integer, and therefore 197 is not a square number.
Anyway, 197 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 197?
The square of a number (here 197) is the result of the product of this number (197) by itself (i.e., 197 × 197); the square of 197 is sometimes called "raising 197 to the power 2", or "197 squared".
As a consequence, 197 is the square root of 38 809.
Number of digits of 197
197 is a number with 3 digits.
What are the multiples of 197?
The multiples of 197 are all integers evenly divisible by 197, that is all numbers such that the remainder of the division by 197 is zero. There are infinitely many multiples of 197. The smallest multiples of 197 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 197 too, since 0 × 197 = 0
- 197: indeed, 197 is a multiple of itself, since 197 is evenly divisible by 197 (we have 197 / 197 = 1, so the remainder of this division is indeed zero)
- 394: indeed, 394 = 197 × 2
- 591: indeed, 591 = 197 × 3
- 788: indeed, 788 = 197 × 4
- 985: indeed, 985 = 197 × 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 197). 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 14.036). 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.