WHAT ARE THE MULTIPLES OF 2? - MATHS - 2025

The multiples of 2 are all even numbers, both positive and negative, not forgetting zero. In a general way it is said that the number "n" is a multiple of "m" if there is an integer "k" such that n = m * k.

So in order to find a multiple of two, m = 2 is substituted and different values ​​are chosen for the integer «k».

For example, if you take m = 2 and k = 5, you get that n = 2 * 5 = 10, that is, 10 is a multiple of 2.

If we take m = 2 and k = -13 we obtain that n = 2 * (- 13) = - 26, therefore 26 is a multiple of 2.

Saying that a number "P" is a multiple of 2 is equivalent to saying that "P" is divisible by 2; that is, when "P" is divided by 2 the result is a whole number.

You may also be interested in what the multiples of 5 are.

What are multiples of 2?

As mentioned above, a number "n" is a multiple of 2 if it has the form n = 2 * k, where "k" is an integer.

It was also mentioned that every even number is a multiple of 2. In order to understand this, the writing of an integer in powers of 10 must be used.

Examples of whole numbers written in powers of 10

If you want to write a number in powers of 10, your writing will have as many addends as there are digits in the number.

The exponents of the powers will depend on the location of each digit.

Some examples are:

- 5 = 5 * (10) ^ 0 = 5 * 1.

- 18 = 1 * (10) ^ 1 + 8 * (10) ^ 0 = 1 * 10 + 8.

- 972 = 9 * (10) ^ 2 + 7 * (10) ^ 1 + 2 * (10) ^ 0 = 9 * 100 + 7 * 10 + 2.

Why are all even numbers multiples of 2?

When decomposing this number into powers of 10, each of the addends that appear, except the last one on the right, is divisible by 2.

To ensure that the number is divisible by 2, all addends must be divisible by 2.

Therefore, the ones digit must be an even number, and if the ones digit is an even number, then the whole number is even.

For this reason, any even number is divisible by 2, and therefore, it is a multiple of 2.

Other approach

If you have a 5-digit number such that it is even, then the number of its units can be written as 2 * k, where «k» is one of the numbers in the set {0, ± 1, ± 2, ± 3, ± 4}.

When decomposing the number into powers of 10, an expression like the following will be obtained:

a * 10,000 + b * 1,000 + c * 100 + d * 10 + e = a * 10,000 + b * 1,000 + c * 100 + d * 10 + 2 * k

By taking the common factor 2 of all the previous expression, it is obtained that the number «abcde» can be written as 2 * (a * 5,000 + b * 500 + c * 50 + d * 5 + k).

Since the expression inside the parentheses is an integer, then it can be concluded that the number "abcde" is a multiple of 2.

In this way you can test for a number with any number of digits, as long as it is even.

Observations

- All negative even numbers are also multiples of 2 and the way to prove it is analogous to what was explained before. The only thing that changes is that a minus sign appears in front of the entire number, but the calculations are the same.

- Zero (0) is also a multiple of 2, since zero can be written as 2 multiplied by zero, that is, 0 = 2 * 0.

References

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7. Tucker, S., & Rambo, J. (2002). Odd and Even Numbers. Capstone.
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