By studying this lesson, you will be able to

• examine whether a whole number is divisible by 3, 4, 6 or 9.

4.1 Examining whether a number is divisible by 3, 4, 6 or 9

It is important to know the divisibility rules when solving problems related to factors and multiples.

If a certain whole number can be divided by another whole number without remainder, then the first number is said to be divisible by the second number. We then identify the second number as a factor of the first number.

6 ÷ 2 = 3 with remainder 0. Therefore, 2 is a factor of 6.

6 ÷ 4 = 1 with remainder 2. Therefore, 4 is not a factor of 6.

One way to find factors of numbers quickly is to use tests of divisibility.

The divisibility rules you learnt in grade 6 are as follows.

• If the ones place digit of a number is divisible by 2, then that number is divisible by 2.

• If the ones place digit of a number is either 0 or 5, then that number is divisible by 5.

• If the ones place digit of a number is 0, then that number is divisible by 10.

• The digital root

Now let us learn about the digital root of a number.

The digital root of a number is calculated by adding up all the digits of that number (and adding the digits of the sums if necessary), until a single digit from 1 to 9 is left. That single digit is defined as the digital root of the relevant number.

Let us see how the digital root of a number is found by considering the following example.

Let us find the digital root of 213. Let us add the digits of 213.

2 + 1 + 3 = 6

Then the digital root of 213 is 6.

The digital root of 242 = 2 + 4 + 2 = 8

Let us find the digital root of 68.

6 + 8 = 14. Let us add the digits of 14. 1 + 4 = 5.

∴ The digital root of 68 is 5.

It is possible to identify certain properties of a number by considering its digital root.

It is possible to identify certain properties of a number by considering its digital root.

• Examining whether a number is divisible by 9

Let us do the following activity. Our goal is to identify a rule to examine whether a number is divisible by 9 or not.

• If the digital root of a whole number is divisible by 9, then that number is divisible by 9. That is 9 is a factor of that number.

• Examining whether a number is divisible by 3

Let us do the following activity. Our goal is to identify a rule to examine whether a number is divisible by 3 or not.

If the digital root of a whole number is divisible by 3, then that number is divisible by 3. That is 3 is a factor of that number.

(1) Without dividing, select the numbers which are divisible by 9.

504" 652" 567" 856" 1143" 1351" 2719" 4536

(2) Without dividing, select and write down the numbers which are divisible by 3.

81, 102, 164, 189, 352, 372, 466, 756, 951, 1029

(3) 3 divides the number 65⬜ . Suggest two digits suitable for the empty

space.

(4) Pencils were brought to be distributed among Nimal’s friends on his birthday Party. The number of pencils was less than 150, but close to it. Nimal observed that each friend could be given 9 pencils. What is the maximum number of pencils that may have been brought?

(5) The following quantities of items were brought to make gift packs to be given to the winners of a competition.

131 exercise books 

130 pencils

128 platignum pens 

131 ballpoint pens

If each gift pack should contain 3 units of each item, write down the minimum extra amounts needed from each item.

• Examining whether a number is divisible by 6

You have learnt previously that, if the ones place digit of a number is zero or an even number, then that number is divisible by 2. You have also learnt how to determine whether a number is divisible by 3. Do the following activity to examine whether a number is divisible by 6.

If a number is divisible by both 2 and 3, then it is divisible by 6.

That is 6 is a factor of that number.

• Examining whether a number is divisible by 4

In order to identify a rule to determine whether a number is divisible by 4, do the following activity.

If the last two digits of a whole number consisting of two or more digits is divisible by 4, then that number is divisible by 4. That is 4 is a factor of that number.

(1) From the following, select and write down the numbers

(i) Which are divisible by 6.

(ii) Which are divisible by 4.

162" 187" 912" 966" 2118" 2123" 2472" 2541" 3024" 3308" 3332" 4800

(2) Write the following numbers in the appropriate column of the table given below. (A number may be written in both column (i) and column (iii).)

348" 496" 288" 414" 1024" 1272" 306" 258" 1008" 6700

(3) The number 62⬜6 is divisible by both 4 and 6. Find the suitable

digit for the empty space.

(4) A drill team arranges themselves in the following manner. On one occasion they form lines consisting of 3 members each and on another occasion lines consisting of 4 members each. They also make circles of 9 members each. If the drill team must have more than 250 members, use the divisibility rules to find the minimum number of members that could be in the team.

(5) Determine whether 126 is divisible by 2, 3, 4, 5, 6, 9 and 10.