It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 821, the answer is: yes, 821 is a prime number because it has only two distinct divisors: 1 and itself (821).
As a consequence, 821 is only a multiple of 1 and 821..
Since 821 is a prime number, 821 is also a deficient number, that is to say 821 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 821 without 821 itself (that is 1, by definition!).
Parity of 821
821 is an odd number, because it is not evenly divisible by 2.
Is 821 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 821 is about 28.653.
Thus, the square root of 821 is not an integer, and therefore 821 is not a square number.
Anyway, 821 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 821?
The square of a number (here 821) is the result of the product of this number (821) by itself (i.e., 821 × 821); the square of 821 is sometimes called "raising 821 to the power 2", or "821 squared".
As a consequence, 821 is the square root of 674 041.
Number of digits of 821
821 is a number with 3 digits.
What are the multiples of 821?
The multiples of 821 are all integers evenly divisible by 821, that is all numbers such that the remainder of the division by 821 is zero. There are infinitely many multiples of 821. The smallest multiples of 821 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 821 too, since 0 × 821 = 0
- 821: indeed, 821 is a multiple of itself, since 821 is evenly divisible by 821 (we have 821 / 821 = 1, so the remainder of this division is indeed zero)
- 1 642: indeed, 1 642 = 821 × 2
- 2 463: indeed, 2 463 = 821 × 3
- 3 284: indeed, 3 284 = 821 × 4
- 4 105: indeed, 4 105 = 821 × 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 821). 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 28.653). 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.