It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 691, the answer is: yes, 691 is a prime number because it has only two distinct divisors: 1 and itself (691).
As a consequence, 691 is only a multiple of 1 and 691..
Since 691 is a prime number, 691 is also a deficient number, that is to say 691 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 691 without 691 itself (that is 1, by definition!).
Parity of 691
691 is an odd number, because it is not evenly divisible by 2.
Is 691 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 691 is about 26.287.
Thus, the square root of 691 is not an integer, and therefore 691 is not a square number.
Anyway, 691 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 691?
The square of a number (here 691) is the result of the product of this number (691) by itself (i.e., 691 × 691); the square of 691 is sometimes called "raising 691 to the power 2", or "691 squared".
As a consequence, 691 is the square root of 477 481.
Number of digits of 691
691 is a number with 3 digits.
What are the multiples of 691?
The multiples of 691 are all integers evenly divisible by 691, that is all numbers such that the remainder of the division by 691 is zero. There are infinitely many multiples of 691. The smallest multiples of 691 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 691 too, since 0 × 691 = 0
- 691: indeed, 691 is a multiple of itself, since 691 is evenly divisible by 691 (we have 691 / 691 = 1, so the remainder of this division is indeed zero)
- 1 382: indeed, 1 382 = 691 × 2
- 2 073: indeed, 2 073 = 691 × 3
- 2 764: indeed, 2 764 = 691 × 4
- 3 455: indeed, 3 455 = 691 × 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 691). 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 26.287). 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.