Is 840 a prime number?

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

For 840, the answer is: No, 840 is not a prime number.

The list of all positive divisors (i.e., the list of all integers that divide 840) is as follows: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840.

For 840 to be a prime number, it would have been required that 840 has only two divisors, i.e., itself and 1.

Find out more:

What is a prime number?

Actually, one can immediately see that 840 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 840 is 0, so it is divisible by 5 and is therefore not prime.

As a consequence:

840 is a multiple of 1
840 is a multiple of 2
840 is a multiple of 3
840 is a multiple of 4
840 is a multiple of 5
840 is a multiple of 6
840 is a multiple of 7
840 is a multiple of 8
840 is a multiple of 10
840 is a multiple of 12
840 is a multiple of 14
840 is a multiple of 15
840 is a multiple of 20
840 is a multiple of 21
840 is a multiple of 24
840 is a multiple of 28
840 is a multiple of 30
840 is a multiple of 35
840 is a multiple of 40
840 is a multiple of 42
840 is a multiple of 56
840 is a multiple of 60
840 is a multiple of 70
840 is a multiple of 84
840 is a multiple of 105
840 is a multiple of 120
840 is a multiple of 140
840 is a multiple of 168
840 is a multiple of 210
840 is a multiple of 280
840 is a multiple of 420

For 840 to be a prime number, it would have been required that 840 has only two divisors, i.e., itself and 1.

Is 840 a deficient number?

No, 840 is not a deficient number: to be deficient, 840 should have been such that 840 is larger than the sum of its proper divisors, i.e., the divisors of 840 without 840 itself (that is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 12 + 14 + 15 + 20 + 21 + 24 + 28 + 30 + 35 + 40 + 42 + 56 + 60 + 70 + 84 + 105 + 120 + 140 + 168 + 210 + 280 + 420 = 2 040).

In fact, 840 is an abundant number; 840 is strictly smaller than the sum of its proper divisors (that is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 12 + 14 + 15 + 20 + 21 + 24 + 28 + 30 + 35 + 40 + 42 + 56 + 60 + 70 + 84 + 105 + 120 + 140 + 168 + 210 + 280 + 420 = 2 040). The smallest abundant number is 12.

Parity of 840

840 is an even number, because it is evenly divisible by 2: 840 / 2 = 420.

Find out more:

What is an even number?

Is 840 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 840 is about 28.983.

Thus, the square root of 840 is not an integer, and therefore 840 is not a square number.

What is the square number of 840?

The square of a number (here 840) is the result of the product of this number (840) by itself (i.e., 840 × 840); the square of 840 is sometimes called "raising 840 to the power 2", or "840 squared".

The square of 840 is 705 600 because 840 × 840 = 840² = 705 600.

As a consequence, 840 is the square root of 705 600.

Number of digits of 840

840 is a number with 3 digits.

What are the multiples of 840?

The multiples of 840 are all integers evenly divisible by 840, that is all numbers such that the remainder of the division by 840 is zero. There are infinitely many multiples of 840. The smallest multiples of 840 are:

0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 840 too, since 0 × 840 = 0
840: indeed, 840 is a multiple of itself, since 840 is evenly divisible by 840 (we have 840 / 840 = 1, so the remainder of this division is indeed zero)
1 680: indeed, 1 680 = 840 × 2
2 520: indeed, 2 520 = 840 × 3
3 360: indeed, 3 360 = 840 × 4
4 200: indeed, 4 200 = 840 × 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 840). 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 28.983). 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 840

Preceding numbers: …838, 839
Following numbers: 841, 842…

Nearest numbers from 840

Preceding prime number: 839
Following prime number: 853