It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 409, the answer is: yes, 409 is a prime number because it has only two distinct divisors: 1 and itself (409).
As a consequence, 409 is only a multiple of 1 and 409..
Since 409 is a prime number, 409 is also a deficient number, that is to say 409 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 409 without 409 itself (that is 1, by definition!).
Parity of 409
409 is an odd number, because it is not evenly divisible by 2.
Is 409 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 409 is about 20.224.
Thus, the square root of 409 is not an integer, and therefore 409 is not a square number.
Anyway, 409 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 409?
The square of a number (here 409) is the result of the product of this number (409) by itself (i.e., 409 × 409); the square of 409 is sometimes called "raising 409 to the power 2", or "409 squared".
As a consequence, 409 is the square root of 167 281.
Number of digits of 409
409 is a number with 3 digits.
What are the multiples of 409?
The multiples of 409 are all integers evenly divisible by 409, that is all numbers such that the remainder of the division by 409 is zero. There are infinitely many multiples of 409. The smallest multiples of 409 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 409 too, since 0 × 409 = 0
- 409: indeed, 409 is a multiple of itself, since 409 is evenly divisible by 409 (we have 409 / 409 = 1, so the remainder of this division is indeed zero)
- 818: indeed, 818 = 409 × 2
- 1 227: indeed, 1 227 = 409 × 3
- 1 636: indeed, 1 636 = 409 × 4
- 2 045: indeed, 2 045 = 409 × 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 409). 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 20.224). 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.