It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 823, the answer is: yes, 823 is a prime number because it has only two distinct divisors: 1 and itself (823).
As a consequence, 823 is only a multiple of 1 and 823..
Since 823 is a prime number, 823 is also a deficient number, that is to say 823 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 823 without 823 itself (that is 1, by definition!).
Parity of 823
823 is an odd number, because it is not evenly divisible by 2.
Is 823 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 823 is about 28.688.
Thus, the square root of 823 is not an integer, and therefore 823 is not a square number.
Anyway, 823 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 823?
The square of a number (here 823) is the result of the product of this number (823) by itself (i.e., 823 × 823); the square of 823 is sometimes called "raising 823 to the power 2", or "823 squared".
As a consequence, 823 is the square root of 677 329.
Number of digits of 823
823 is a number with 3 digits.
What are the multiples of 823?
The multiples of 823 are all integers evenly divisible by 823, that is all numbers such that the remainder of the division by 823 is zero. There are infinitely many multiples of 823. The smallest multiples of 823 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 823 too, since 0 × 823 = 0
- 823: indeed, 823 is a multiple of itself, since 823 is evenly divisible by 823 (we have 823 / 823 = 1, so the remainder of this division is indeed zero)
- 1 646: indeed, 1 646 = 823 × 2
- 2 469: indeed, 2 469 = 823 × 3
- 3 292: indeed, 3 292 = 823 × 4
- 4 115: indeed, 4 115 = 823 × 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 823). 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 28.688). 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.