It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 653, the answer is: yes, 653 is a prime number because it has only two distinct divisors: 1 and itself (653).
As a consequence, 653 is only a multiple of 1 and 653..
Since 653 is a prime number, 653 is also a deficient number, that is to say 653 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 653 without 653 itself (that is 1, by definition!).
Parity of 653
653 is an odd number, because it is not evenly divisible by 2.
Is 653 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 653 is about 25.554.
Thus, the square root of 653 is not an integer, and therefore 653 is not a square number.
Anyway, 653 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 653?
The square of a number (here 653) is the result of the product of this number (653) by itself (i.e., 653 × 653); the square of 653 is sometimes called "raising 653 to the power 2", or "653 squared".
As a consequence, 653 is the square root of 426 409.
Number of digits of 653
653 is a number with 3 digits.
What are the multiples of 653?
The multiples of 653 are all integers evenly divisible by 653, that is all numbers such that the remainder of the division by 653 is zero. There are infinitely many multiples of 653. The smallest multiples of 653 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 653 too, since 0 × 653 = 0
- 653: indeed, 653 is a multiple of itself, since 653 is evenly divisible by 653 (we have 653 / 653 = 1, so the remainder of this division is indeed zero)
- 1 306: indeed, 1 306 = 653 × 2
- 1 959: indeed, 1 959 = 653 × 3
- 2 612: indeed, 2 612 = 653 × 4
- 3 265: indeed, 3 265 = 653 × 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 653). 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 25.554). 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.