Is 420 a prime number?

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

For 420, the answer is: No, 420 is not a prime number.

The list of all positive divisors (i.e., the list of all integers that divide 420) is as follows: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420.

For 420 to be a prime number, it would have been required that 420 has only two divisors, i.e., itself and 1.

Find out more:

What is a prime number?

Actually, one can immediately see that 420 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 420 is 0, so it is divisible by 5 and is therefore not prime.

As a consequence:

420 is a multiple of 1
420 is a multiple of 2
420 is a multiple of 3
420 is a multiple of 4
420 is a multiple of 5
420 is a multiple of 6
420 is a multiple of 7
420 is a multiple of 10
420 is a multiple of 12
420 is a multiple of 14
420 is a multiple of 15
420 is a multiple of 20
420 is a multiple of 21
420 is a multiple of 28
420 is a multiple of 30
420 is a multiple of 35
420 is a multiple of 42
420 is a multiple of 60
420 is a multiple of 70
420 is a multiple of 84
420 is a multiple of 105
420 is a multiple of 140
420 is a multiple of 210

For 420 to be a prime number, it would have been required that 420 has only two divisors, i.e., itself and 1.

Is 420 a deficient number?

No, 420 is not a deficient number: to be deficient, 420 should have been such that 420 is larger than the sum of its proper divisors, i.e., the divisors of 420 without 420 itself (that is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 10 + 12 + 14 + 15 + 20 + 21 + 28 + 30 + 35 + 42 + 60 + 70 + 84 + 105 + 140 + 210 = 924).

In fact, 420 is an abundant number; 420 is strictly smaller than the sum of its proper divisors (that is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 10 + 12 + 14 + 15 + 20 + 21 + 28 + 30 + 35 + 42 + 60 + 70 + 84 + 105 + 140 + 210 = 924). The smallest abundant number is 12.

Parity of 420

420 is an even number, because it is evenly divisible by 2: 420 / 2 = 210.

Find out more:

What is an even number?

Is 420 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 420 is about 20.494.

Thus, the square root of 420 is not an integer, and therefore 420 is not a square number.

What is the square number of 420?

The square of a number (here 420) is the result of the product of this number (420) by itself (i.e., 420 × 420); the square of 420 is sometimes called "raising 420 to the power 2", or "420 squared".

The square of 420 is 176 400 because 420 × 420 = 420² = 176 400.

As a consequence, 420 is the square root of 176 400.

Number of digits of 420

420 is a number with 3 digits.

What are the multiples of 420?

The multiples of 420 are all integers evenly divisible by 420, that is all numbers such that the remainder of the division by 420 is zero. There are infinitely many multiples of 420. The smallest multiples of 420 are:

0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 420 too, since 0 × 420 = 0
420: indeed, 420 is a multiple of itself, since 420 is evenly divisible by 420 (we have 420 / 420 = 1, so the remainder of this division is indeed zero)
840: indeed, 840 = 420 × 2
1 260: indeed, 1 260 = 420 × 3
1 680: indeed, 1 680 = 420 × 4
2 100: indeed, 2 100 = 420 × 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 420). 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 20.494). 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 420

Preceding numbers: …418, 419
Following numbers: 421, 422…

Nearest numbers from 420

Preceding prime number: 419
Following prime number: 421