Understanding The Rule of Divisibility by 3: A Dialogue Approach by Paul Chika Emekwulu

This article is a continuation of a previous article on divisibility rule of whole numbers. Here, in this article we are particularly investigating the divisibility rule for 3.

Introduction

The students featured in this dialogue are Amina, Adedeji, Adeyemi and Ajalla.  The school and the teacher are the same as in the previous dialogue.  So, the teacher is still Mrs. Adebayo Adeyinka while the school is still Women’s National Teachers’ Institute, Gboko, Benue state.

Objectives

At the end of the dialogue, the students should be able to:

(a)  build the rule for divisibility by 3.
(b)  state the rule for divisibility by 3.
(c)  answer questions based on the rule for divisibility by 3.
(d)  search for counter examples.

Previous Knowledge or Prerequisite

The students are familiar with digits of numbers.

The Dialogue Begins

Mrs. Adeyinka:  “The next rule on our list is that of 3.  We need someone to tell us what the rule says ?”

Amina:  “A number is divisible by 3 is the sum of its digit is divisible by 3 e.g. 363.

3 + 6 + 3 =12 and 12 is divisible by 3.

Therefore, 363 is divisible by 3.”

Mrs. Adeyinka:  “You are very correct but why is this so ?”

(There was a long silence.)

363 can be expressed as: 300+63.

=(3×100)+(6×10)+(3×1)

=3(100)+6(10)+3(1)

=3(100-1)+3x[6(10-1)]+6+(3×1))

=3(99) + 3+6(9) + 6+3

=3(99)+6(9)+(3+6+3)

Now 3 is a factor of 3(99) and 6(9). 

Why do you think that 3 is a factor of 6(9) ?”

Adedeji:  “This is because 6 is a multiple of 2.

6(9) can be expressed as 3(18).”

Mrs. Adeyinka:  “What again do you observe ?”

Adedeji:  “3 is a also a factor of 3(99).”

Mrs. Adeyinka:  “Do you agree with Adedeji, Ajalla ?”

Ajalla:  “Yes, I do.”

Mrs. Adeyinka:  “Amina wants to say something.

What do you want to say? Amina.”

Amina:  “Do you mean that 6(9) being a multiple of 3 implies that 3 is a factor of 6(9) ?”

Mrs. Adeyinka:  “Let Adeyemi answer your question, Ajalla.

What do you say? Adeyemi.”

Adeyemi:  “I believe that is true.  This is because 6(9) being a multiple of 3 means that 6(9) is divisible by 3.  The number 6(9) after all is equal to 54 and 18×3=54.”

Mrs. Adeyinka:  “This is true.”

Amina:  “I have a question, Mrs. Adeyinka.  We have said that 3(99) and 6(9) are both multiples of 3 and are therefore, divisible by 3.

What of 3+6+3 ?.”

Mrs. Adeyinka:  “What do you think yourself? Amina.”

Amina:  “3+6+3=12 and 12 is divisible by 3, can we then draw a conclusion that 363 is divisible by 3 ?”

Mrs. Adeyinka:  “Yes.  There is one more thing.

An observation?  An observation. A beautiful observation.  An interesting idea. Who has noticed what I am talking about ?

(There was silence.}

We found the answers to 3+6+3 maybe without much importance attached to these numbers being added.

To me I think there is more to this.

The digits 3, 6, 3 are the very digits of the number 363 and I feel this is not a coincidence.

Assuming your observation is true what meaning does it have for divisibility rule by 3 ?

Adeyemi:  “No idea.”

Mrs. Adeyinka:  “Let us investigate your ideas. Investigation is always a wonderful way to find out more.

124=(1×100) + (2×10)+(4×1)

=1(100) = 2(10) + 4(1)

=1(100-1) + 1×2(10-1)+(1+2)+4(1)

=1(99)+1×2(9)+(1+2)+4(1)

=1(99)+2(9)+(1+2)

What again do we notice ?”

Ajalla:  “1(99), 2(9) are divisible by 3 but 1+2+4 is not.”

Mrs.  Adeyinka:  “Do you think that 124 is divisible by 3 ?”

Ajalla:  “No, it is not.”

Mrs. Adeyinka:  “Why ?”

Ajalla:  “Maybe because 1+2+4 is not divisible by 3.”

Mrs. Adeyinka:  “Do you agree with me that 1, 2, 4, are the digits of our original number 124?”

Amina & others:  “Yes, we do.”

Mrs. Adeyinka:  “Do you agree that if c=a+b, and if a and b are both divisible by 3, then c is divisible by 3 ?”

Amina & others:  “Yes, we do.

What more does this mean for us ?  “Anybody !”

Amina:  “This means that the sum of two or more multiples of 3 is also a multiple of 3.”

Mrs. Adeyinka:  “Okay.  I think that from what we have said so far, we can now draw a conclusion about the divisibility rule by 3.”

Adeyemi:  (Shouting and rejoicing)  “Yes, we can.

A number is divisible by 3 if the sum of its digits is divisible by 3.”

Mrs. Adeyinka:  (talking to Ajalla) “Do you agree with Adeyemi, or do you have anything different to say ?”

Ajalla:  “No, I don’t have anything different.”

Mrs. Adeyinka:  “Then, you agree with Adeyemi.”

Ajalla: “Yes, I do.”

Mrs. Adeyinka:  “I am thanking all of us for an investigation well done.”

EXAMPLE 2:

Find the smallest number that can be added to 421 to make the resulting number divisible by 3.
 
SOLUTION

421 + 2 = 423

421 + (2+3) = 421 + 5 = 426

421+(5+3) = 421+ 8 = 429

421+(5+3)+3 = 421+11= 432

421+(5+3+3)+3 = 421+14 = 435

421+(5+3+3+3) + 3 = 421+17 = 438

421+(5+3+3+3+3) + 3 = 421 + 20 = 441

The replacement set is 2, 5, 8, 11, 14, 17, 20, …

Therefore, 2 is the smallest number to be added to 421 to make the resulting number divisible by 3.

Verifying Our Result

This can be verified algebraically thus.

Let the smallest number to be added to 421 be x.

(4+2+1+x)/3 = 9/3

7 + x = 9.  From here, x = 9 – 7 = 2.

Therefore, the smallest number to be added to 421 to make the resulting number divisible by 3 is 2.

A Search for Counter Examples

(a) Can you identify any number p whose sum of digits is divisible by 3, but itself is not divisible by 3 ?

(b) Conversely, can you think of a number p divisible by 3 but whose sum of digits is not divisible by 3 ?

Chapter Summary

(a)  If c=a+b and a, b, are divisible by 3, then c is also divisible by 3.

(b)  A number is divisible by 3 if the sum of its digits is divisible by 3.

e.g. 36789.

13 + (2+0) + 3 + 2 = 15

13+(2+3+3+5= 18

13 + (5+3) = 13+8= 21

13+(8+3)= 13+11= 24

13 + (11+3) = 13 + 14 = 27

13+(14+3)=13 + 17 = 30

The smallest number is 2.

Therefore, the smallest number to be added to 13 to make the resulting number divisible by 3 is 2.

When you add 17 to 13, the result is 30 and 30 is divisible by 3.

When you add 14 to 13, the sum is 27, and 27 is divisible by 3.

When you add 11 to 13, the sum is 24, and 24 is divisible by 3.

When you add 8 to 13, the sum is 21 and 21 is divisible by 3.

When you add 5 to 13, the sum is 18 and 18 is divisible by 3.

When you add 2 to 13, the sum is 15 and 15 is divisible by 3.

(a)  A number p is divisible by 3 if the sum of digits in $ is also divisible by 3.

(b)  Any three digit number can be expressed as:

abc and abc = 100a + 10b + c.

Oral Exercises

1.  Answer true or false to the following questions:

(a)  Divisibility rule by 3 says that a number p is divisible by 3 if the sum of digits in p is divisible by 9.

(b) Divisibility rule by 3 says that a number p is divisible by 3 if the sum of digits in p is divisible by 3.

2.  State the divisibility rule by 3.

Written Exercises

1.  Which of these numbers are divisible by 3?  Circle them.

(a)  4653
(b)  4765
(c)  6847
(d)  9876
(e)  3483
(f)  6549

(g)  6539
(h)  7650
(i)  8675
(j)  9828
(k)  7654
(l)  4356

(m)  6598
(n)  6874
(o)  6789
(p)  5467
(q)  4653
(r)  6547

2.  Find the smallest number needed to make 358 divisible by 3.

3. If 5674$M$5 is divisible by 3, what is M?

4.  At the end – of – the – year school party, a high school Principal has a total of 4372 apples.  Each student gets 3 apples.

(a)  How many apples does the Principal need ?
(b)  How many students are there in the school ?
(c) How much did he spend on the apples if 1 apple costs 60 naira ?

5. Find the smallest number that can be added to 682 to make the resulting number divisible by 3.

6   Find the smallest number that can be added to 721 to make the resulting number divisible by 3.

7. Find the smallest number that can be added to 235 to make the resulting number divisible by 3.

8.  How many numbers divisible by 3 are there between 3 and 36 ?

9.  How many numbers divisible by 3 are there between 12 and 144 ?

10.  How many multiples of 3 are there between 12 and 400 ?

11.  The sum of two multiples of 3 is 27.  The difference between twice the smaller and the larger number is 9.  Find the numbers.

Mrs. Adeyinka:  “Next on to divisibility rule by 4.  My name is Mrs. Adeyinka. Good evening everyone.”

The Dialogue Ends

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