Is 400 a prime number?

It is possible to find out using mathematical methods whether a given integer is a prime number or not.

For 400, the answer is: No, 400 is not a prime number.

The list of all positive divisors (i.e., the list of all integers that divide 400) is as follows: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400.

For 400 to be a prime number, it would have been required that 400 has only two divisors, i.e., itself and 1.

Find out more:

What is a prime number?

Actually, one can immediately see that 400 cannot be prime, because 5 is one of its divisors: indeed, a number ending with 0 or 5 has necessarily 5 among its divisors. The last digit of 400 is 0, so it is divisible by 5 and is therefore not prime.

As a consequence:

400 is a multiple of 1
400 is a multiple of 2
400 is a multiple of 4
400 is a multiple of 5
400 is a multiple of 8
400 is a multiple of 10
400 is a multiple of 16
400 is a multiple of 20
400 is a multiple of 25
400 is a multiple of 40
400 is a multiple of 50
400 is a multiple of 80
400 is a multiple of 100
400 is a multiple of 200

For 400 to be a prime number, it would have been required that 400 has only two divisors, i.e., itself and 1.

Is 400 a deficient number?

No, 400 is not a deficient number: to be deficient, 400 should have been such that 400 is larger than the sum of its proper divisors, i.e., the divisors of 400 without 400 itself (that is 1 + 2 + 4 + 5 + 8 + 10 + 16 + 20 + 25 + 40 + 50 + 80 + 100 + 200 = 561).

In fact, 400 is an abundant number; 400 is strictly smaller than the sum of its proper divisors (that is 1 + 2 + 4 + 5 + 8 + 10 + 16 + 20 + 25 + 40 + 50 + 80 + 100 + 200 = 561). The smallest abundant number is 12.

Parity of 400

400 is an even number, because it is evenly divisible by 2: 400 / 2 = 200.

Find out more:

What is an even number?

Is 400 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 400 is 20.

Therefore, the square root of 400 is an integer, and as a consequence 400 is a perfect square.

As a consequence, 20 is the square root of 400.

What is the square number of 400?

The square of a number (here 400) is the result of the product of this number (400) by itself (i.e., 400 × 400); the square of 400 is sometimes called "raising 400 to the power 2", or "400 squared".

The square of 400 is 160 000 because 400 × 400 = 400² = 160 000.

As a consequence, 400 is the square root of 160 000.

Number of digits of 400

400 is a number with 3 digits.

What are the multiples of 400?

The multiples of 400 are all integers evenly divisible by 400, that is all numbers such that the remainder of the division by 400 is zero. There are infinitely many multiples of 400. The smallest multiples of 400 are:

0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 400 too, since 0 × 400 = 0
400: indeed, 400 is a multiple of itself, since 400 is evenly divisible by 400 (we have 400 / 400 = 1, so the remainder of this division is indeed zero)
800: indeed, 800 = 400 × 2
1 200: indeed, 1 200 = 400 × 3
1 600: indeed, 1 600 = 400 × 4
2 000: indeed, 2 000 = 400 × 5
etc.

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 400). 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 20). 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.

Numbers near 400

Preceding numbers: …398, 399
Following numbers: 401, 402…

Nearest numbers from 400

Preceding prime number: 397
Following prime number: 401