It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 401, the answer is: yes, 401 is a prime number because it has only two distinct divisors: 1 and itself (401).
As a consequence, 401 is only a multiple of 1 and 401..
Since 401 is a prime number, 401 is also a deficient number, that is to say 401 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 401 without 401 itself (that is 1, by definition!).
Parity of 401
401 is an odd number, because it is not evenly divisible by 2.
Is 401 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 401 is about 20.025.
Thus, the square root of 401 is not an integer, and therefore 401 is not a square number.
Anyway, 401 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 401?
The square of a number (here 401) is the result of the product of this number (401) by itself (i.e., 401 × 401); the square of 401 is sometimes called "raising 401 to the power 2", or "401 squared".
As a consequence, 401 is the square root of 160 801.
Number of digits of 401
401 is a number with 3 digits.
What are the multiples of 401?
The multiples of 401 are all integers evenly divisible by 401, that is all numbers such that the remainder of the division by 401 is zero. There are infinitely many multiples of 401. The smallest multiples of 401 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 401 too, since 0 × 401 = 0
- 401: indeed, 401 is a multiple of itself, since 401 is evenly divisible by 401 (we have 401 / 401 = 1, so the remainder of this division is indeed zero)
- 802: indeed, 802 = 401 × 2
- 1 203: indeed, 1 203 = 401 × 3
- 1 604: indeed, 1 604 = 401 × 4
- 2 005: indeed, 2 005 = 401 × 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 401). 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 20.025). 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.