It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 857, the answer is: yes, 857 is a prime number because it has only two distinct divisors: 1 and itself (857).
As a consequence, 857 is only a multiple of 1 and 857..
Since 857 is a prime number, 857 is also a deficient number, that is to say 857 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 857 without 857 itself (that is 1, by definition!).
Parity of 857
857 is an odd number, because it is not evenly divisible by 2.
Is 857 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 857 is about 29.275.
Thus, the square root of 857 is not an integer, and therefore 857 is not a square number.
Anyway, 857 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 857?
The square of a number (here 857) is the result of the product of this number (857) by itself (i.e., 857 × 857); the square of 857 is sometimes called "raising 857 to the power 2", or "857 squared".
As a consequence, 857 is the square root of 734 449.
Number of digits of 857
857 is a number with 3 digits.
What are the multiples of 857?
The multiples of 857 are all integers evenly divisible by 857, that is all numbers such that the remainder of the division by 857 is zero. There are infinitely many multiples of 857. The smallest multiples of 857 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 857 too, since 0 × 857 = 0
- 857: indeed, 857 is a multiple of itself, since 857 is evenly divisible by 857 (we have 857 / 857 = 1, so the remainder of this division is indeed zero)
- 1 714: indeed, 1 714 = 857 × 2
- 2 571: indeed, 2 571 = 857 × 3
- 3 428: indeed, 3 428 = 857 × 4
- 4 285: indeed, 4 285 = 857 × 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 857). 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 29.275). 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.