We examine more about divisibility rules and see why they work. In this post, we discuss divisibility by 4. If you can recall, divisibility by 4 states that
A number is divisible by 4 if the last two digits are divisible by 4.
The last two digits in that rule means the tens and ones digits.
Now, why do we only should consider the last two digits? Why is it that 76200348 divisible by 4 just because 48 is divisible by 4.
If we examine the structure of numbers, every whole number can be broken down into the representation
For example, 325 can be represented and can be represented as
Now, here is the trick. Since 100 is divisible by 4, then a(100) part is always divisible by 4. That is why, we only have to check the last two digits. Continue Reading →
In the previous post, we have discussed about divisibility by 2. In this post, we discuss about divisibility by 3.
Rule: A number is divisible by 3 if the sum of the digits is divisible by 3.
The number 321 is divisible by 3 because 3 + 2 + 1 = 6 is divisible by 3. On the other hand, the number 185 is not divisible by 3 because 1 + 8 + 5 = 14 is not divisible by 3. Now, why does this rule work?
Notice how the numbers are represented in expanded notation:
This means that number in hundreds can be represented as
where h, t, u are the hundreds, tens, and units digits. Now, we can represent as
and regroup the terms as
Of course, is divisible by 3, so it only remains to show that is divisible by 3. But, is the sum of the digits of a 3-digit number. This proves (for three digit numbers) that the rule above is true.
Although the proof above works only for 3-digit numbers, it can be done to any number of digits.
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.
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. Continue Reading →