WHAT ARE THE MULTIPLES OF 8? - MATHS - 2024

The multiples of 8 are all the numbers that result from multiplying 8 by another whole number. To identify what the multiples of 8 are, it is necessary to know what it means for one number to be a multiple of another.

An integer "n" is said to be a multiple of the integer "m" if there is an integer "k", such that n = m * k.

So to know if a number "n" is a multiple of 8, we must substitute m = 8 in the previous equality. Therefore, we obtain n = 8 * k.

That is, the multiples of 8 are all those numbers that can be written as 8 multiplied by some whole number. For example:

- 8 = 8 * 1, so 8 is a multiple of 8.

- -24 = 8 * (- 3). That is, -24 is a multiple of 8.

What are the multiples of 8?

The Euclidean division algorithm says that given two integers "a" and "b" with b ≠ 0, there are only integers "q" and "r", such that a = b * q + r, where 0≤ r <-b-.

When r = 0 it is said that "b" divides "a"; that is, "a" is divisible by "b".

If b = 8 and r = 0 are substituted in the division algorithm, we get that a = 8 * q. That is, the numbers that are divisible by 8 have the form 8 * q, where "q" is an integer.

How to know if a number is a multiple of 8?

We already know that the form of the numbers that are multiples of 8 is 8 * k, where "k" is an integer. Rewriting this expression you can see that:

8 * k = 2³ * k = 2 * (4 * k)

With this last way of writing the multiples of 8, it is concluded that all the multiples of 8 are even numbers, with which all the odd numbers are discarded.

The expression "2³ * k" indicates that for a number to be a multiple of 8 it must be divisible 3 times by 2.

That is, when dividing the number "n" by 2, a result "n1" is obtained, which in turn is divisible by 2; and that after dividing «n1» by 2 we obtain a result «n2», which is also divisible by 2.

Example

Dividing the number 16 by 2 gives the result 8 (n1 = 8). When 8 is divided by 2 the result is 4 (n2 = 4). And finally, when 4 is divided by 2, the result is 2.

So 16 is a multiple of 8.

On the other hand, the expression "2 * (4 * k)" implies that, for a number to be a multiple of 8, it must be divisible by 2 and then by 4; that is, when dividing the number by 2, the result is divisible by 4.

Example

Dividing the number -24 by 2 gives a result of -12. And by dividing -12 by 4 the result is -3.

Therefore, the number -24 is a multiple of 8.

Some multiples of 8 are: 0, ± 8, ± 16, ± 32, ± 40, ± 48, ± 56, ± 64, ± 72, ± 80, ± 88, ± 96, and more.

Observations

- Euclid's division algorithm is written for whole numbers, so multiples of 8 are both positive and negative.

- The number of numbers that are multiples of 8 is infinite.

References

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