It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 421, the answer is: yes, 421 is a prime number because it has only two distinct divisors: 1 and itself (421).
As a consequence, 421 is only a multiple of 1 and 421..
Since 421 is a prime number, 421 is also a deficient number, that is to say 421 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 421 without 421 itself (that is 1, by definition!).
Parity of 421
421 is an odd number, because it is not evenly divisible by 2.
Is 421 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 421 is about 20.518.
Thus, the square root of 421 is not an integer, and therefore 421 is not a square number.
Anyway, 421 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 421?
The square of a number (here 421) is the result of the product of this number (421) by itself (i.e., 421 × 421); the square of 421 is sometimes called "raising 421 to the power 2", or "421 squared".
As a consequence, 421 is the square root of 177 241.
Number of digits of 421
421 is a number with 3 digits.
What are the multiples of 421?
The multiples of 421 are all integers evenly divisible by 421, that is all numbers such that the remainder of the division by 421 is zero. There are infinitely many multiples of 421. The smallest multiples of 421 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 421 too, since 0 × 421 = 0
- 421: indeed, 421 is a multiple of itself, since 421 is evenly divisible by 421 (we have 421 / 421 = 1, so the remainder of this division is indeed zero)
- 842: indeed, 842 = 421 × 2
- 1 263: indeed, 1 263 = 421 × 3
- 1 684: indeed, 1 684 = 421 × 4
- 2 105: indeed, 2 105 = 421 × 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 421). 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 20.518). 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.