It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 827, the answer is: yes, 827 is a prime number because it has only two distinct divisors: 1 and itself (827).
As a consequence, 827 is only a multiple of 1 and 827..
Since 827 is a prime number, 827 is also a deficient number, that is to say 827 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 827 without 827 itself (that is 1, by definition!).
Parity of 827
827 is an odd number, because it is not evenly divisible by 2.
Is 827 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 827 is about 28.758.
Thus, the square root of 827 is not an integer, and therefore 827 is not a square number.
Anyway, 827 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 827?
The square of a number (here 827) is the result of the product of this number (827) by itself (i.e., 827 × 827); the square of 827 is sometimes called "raising 827 to the power 2", or "827 squared".
As a consequence, 827 is the square root of 683 929.
Number of digits of 827
827 is a number with 3 digits.
What are the multiples of 827?
The multiples of 827 are all integers evenly divisible by 827, that is all numbers such that the remainder of the division by 827 is zero. There are infinitely many multiples of 827. The smallest multiples of 827 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 827 too, since 0 × 827 = 0
- 827: indeed, 827 is a multiple of itself, since 827 is evenly divisible by 827 (we have 827 / 827 = 1, so the remainder of this division is indeed zero)
- 1 654: indeed, 1 654 = 827 × 2
- 2 481: indeed, 2 481 = 827 × 3
- 3 308: indeed, 3 308 = 827 × 4
- 4 135: indeed, 4 135 = 827 × 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 827). 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 28.758). 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.