To find out what the divisors of 8 are, as well as any other integer, we start by doing a prime factorization. It is a fairly short process and easy to learn.
When speaking of prime factorization, we are referring to two definitions: factors and prime numbers.
Prime numbers are those natural numbers that are divisible only by the number 1 and by themselves.
Decomposing a whole number into prime factors refers to rewriting that number as a product of prime numbers, where each is called a factor.
For example, 6 can be written as 2 * 3; therefore 2 and 3 are the prime factors in the decomposition.
Divisors of 8
The divisors of 8 are all those integers that, when dividing 8 between them, the result is also an integer less than 8.
Another way to define them is as follows: an integer "m" is a divisor of 8 if when dividing 8 by "m" (8 ÷ m), the remainder or remainder of said division is equal to 0.
The decomposition of a number into prime factors is obtained by dividing the number by the prime numbers smaller than this.
To determine what the divisors of 8 are, first the number 8 is decomposed into prime factors, where it is obtained that 8 = 2³ = 2 * 2 * 2.
The above indicates that the only prime factor that 8 has is 2, but this is repeated 3 times.
How are the divisors obtained?
Having done the decomposition into prime factors, we proceed to calculate all the possible products between said prime factors.
In the case of 8, there is only one prime factor that is 2, but it is repeated 3 times. Therefore, the divisors of 8 are: 2, 2 * 2 and 2 * 2 * 2. That is: {2, 4, 8}.
To the previous list it is necessary to add the number 1, since 1 is always a divisor of any whole number. Therefore, the list of divisors of 8 so far is: {1, 2, 4, 8}.
Are there more dividers?
The answer to this question is yes. But which divisors are missing?
As said before, all the divisors of a number are the possible products between the prime factors of that number.
But it was also indicated that the divisors of 8 are all those integers, such that when dividing 8 between them the remainder of the division is equal to 0.
The last definition speaks of integers in a general way, not just positive integers. Therefore, you also need to add the negative integers that divide 8.
The negative integers that divide 8 are the same as those found above, with the difference that the sign will be negative. That is, -1, -2, -4 and -8 must be added.
With what has been said before, it is concluded that all the divisors of 8 are: {± 1, ± 2, ± 4, ± 8}.
Observation
The definition of divisors of a number is limited only to integers. Otherwise, it could also be said that 1/2 divides 8, since when dividing between 1/2 and 8 (8 ÷ 1/2), the result is 16, which is an integer.
The method presented in this article to find the divisors of the number 8 can be applied to any whole number.
References
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