It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 523, the answer is: yes, 523 is a prime number because it has only two distinct divisors: 1 and itself (523).
As a consequence, 523 is only a multiple of 1 and 523..
Since 523 is a prime number, 523 is also a deficient number, that is to say 523 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 523 without 523 itself (that is 1, by definition!).
Parity of 523
523 is an odd number, because it is not evenly divisible by 2.
Is 523 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 523 is about 22.869.
Thus, the square root of 523 is not an integer, and therefore 523 is not a square number.
Anyway, 523 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 523?
The square of a number (here 523) is the result of the product of this number (523) by itself (i.e., 523 × 523); the square of 523 is sometimes called "raising 523 to the power 2", or "523 squared".
As a consequence, 523 is the square root of 273 529.
Number of digits of 523
523 is a number with 3 digits.
What are the multiples of 523?
The multiples of 523 are all integers evenly divisible by 523, that is all numbers such that the remainder of the division by 523 is zero. There are infinitely many multiples of 523. The smallest multiples of 523 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 523 too, since 0 × 523 = 0
- 523: indeed, 523 is a multiple of itself, since 523 is evenly divisible by 523 (we have 523 / 523 = 1, so the remainder of this division is indeed zero)
- 1 046: indeed, 1 046 = 523 × 2
- 1 569: indeed, 1 569 = 523 × 3
- 2 092: indeed, 2 092 = 523 × 4
- 2 615: indeed, 2 615 = 523 × 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 523). 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 22.869). 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.