The Explanation Behind Divisibility by 2

One of the basic mathematics that we learn in middle school mathematics is divisibility. There are rules that are usually memorized. A number is divisible by 2 if it is even; a number is divisible by 3 if the sum of its digits is divisible by 3. In this series, we are going to discuss the proof or the explanations behind these rules. We start with divisibility by 2.

Rule: If a number is even, then it is divisible by 2.

Explanation

1-digit numbers

A number is even if it ends in 0, 2, 4, 6, and 8. Now, if we are talking only about 1-digit numbers, then we are sure that all of them are divisible by 2.

2-digit numbers and above

For 2-digit numbers and above, we can generalize by the following representation. Notice that any number greater than 10 can be represented by multiplying some number by 10 and adding the one’s digit. Here are a few examples for even numbers.

  • 12 = 1(10) + 2
  • 34 = 3(10) + 4
  • 186 = 18 (10) + 6
  • 2388 = 238(10) + 8
  • 290 = 29 (10) + 0

From these statements, all even numbers can be represented as 10 times some number n, and then add the even ones digit.

  • 10n
  • 10n + 2
  • 10n + 4
  • 10n + 6
  • 10n + 8

Notice that using these representations, we can represent all even numbers. This means that if we can show that we can divide these five representations by 2, we know that all even numbers are divisible by 2. But I guess it is already obvious by now, that all the representations above are divisible by 2: 10n divided by 2 is 5n, and 0, 2, 4, 6, and 8 are all divisible by 2. In other words, we can always factor out 2. For example, 10n + 6 = 2(5n + 3). This means that 2 is a multiple of 10n + 6 and it is therefore divisible by 2.

Note: I know this is obvious, but this post is written for middle school students.

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