Divisibility Rules

Division also requires fluency with division facts. One way to achieve this is by studying the multiplication tables, since division & multiplication are mirror images of each other.

Using the One Page Multiplication Table

We have seen earlier, the construction of the one-page multiplication table. If any number, say 42 appears in the table then we can immediately tell its factors; 6 & 7 in this case. Hence, by repeatedly referring to the table, division facts of many numbers can be remembered.

Divisibility Rules

For other numbers, it helps to understand the divisibility rules of various numbers. These are rules which help us to decide the number which can divide a given number. Our intention here is not to just give the rules but enable understanding of why the rules work. Here the visualization of a number in terms of bundles & sticks or Sheets, strips & pieces would be very helpful.

We also need to keep in mind that these rules work only if the number is represented with the decimal system.

By 2

In a number the bundles or sheets or strips are always even since in the decimal system they are made of ten sticks. Ten is divisible by 2. Hence, we need only to check whether the number in the Unit’s place is divisible by 2 i.e whether they are 2,4, 6 0r 8.

We can add 0 to the above list since even if they are no sticks, the number which consists only of bundles will be divisible by 2.

By 4

In a 3-digit number, the sheet being hundred is divisible by 4. But the bundle being ten is not. Hence, we need to see if the 2 digit number formed from the numerals in the ten’s and unit’s place is divisible by 4.

By 5

Bundles, sheets & strips are divisible by 5. Hence if a number has 5 sticks or no sticks, it will be divisible by 5. Hence any number ending with 0 or 5 will be divisible by 5.

By 3 or 9

Let us think of a number formed with numerals A, B& C. It would be written as ABC and can be represented by A sheets, B strips & C pieces.

From each sheet if we remove 1 piece, then it becomes 99 which is divisible by 3 or 9. Hence for divisibility purposes, we need not be concerned with them. We would be left with A pieces, one from each of the A sheets.

When we remove 1 piece from each strip, it would become 9 which again is divisible by 3 or 9. Again for divisibility purposes, we need not be concerned with hem. We would be left with B pieces, one from each of the strips.

We already have C pieces of the original number.

Hence, we have a total of A+B+C pieces, which is the number A+B+C since each piece equals 1. Our problem has been reduced to finding out if A+B+C is divisible by either 3 or 9. A+B+C is the sum of the digits in the original number.

If the sum of the digits is divisible by 3, then the number is divisible y 3.

If the sum of the digits is divisible by 9, then the number is divisible by 9. It is also divisible by 3 since 9 itself is divisible by 3.

We need to remember that there would be cases where the sum of the digits is divisible by 3 but not by 7. A+B+C being 15 is an example.

By 6

A number divisible by 6 should be divisible by both 2 and 3 which are factors of 6.

By 8

By the same logic, a number divisible by 8 should be divisible by both 2 and 4.

In numbers greater than 999, since one thousand is divisible by 8, we need only to check if the number formed by the digits in the hundred’s, ten’s and unit’s place is divisible by 8.

By 7

The logic for explaining the divisibility rule for 7 is quite complicated. But most people are not even aware of the rule. So, we will just explain the rule.

Assume that the number is “abcdefgh” where each letter stands for a digit. There is no limitation on the number of digits in the number. The rule applies even in cases where some of the digits are repeated.

1. Split the number into 2 parts

2. The first part represents the last 3 digits, in this case “fgh”.

3. The second part represents the rest of the digits, in this case “abcde”.

4. Find the difference between the above 2 numbers, say it is “pqrst”

5. Repeat the steps (a) to (d) with the new number “pqrst”.

6. Repeat the process until the resulting number is a 2-digit number.

7. Check whether the 2-digit number is divisible by 7.

8. If yes, then the starting number “abcdefgh” is also divisible by 7.

9. If not, then the starting number “abcdefgh” is not divisible by 7.

In step (d) we have used the term difference. This is because in later stages the “last 3 digits” may represent a bigger number than the remaining digits.