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The Test of Divisibility of 3
We use the numbers 1, 2, 3 and 4 ... for counting. These numbers are called natural numbers.
All natural numbers together with 0 are called whole numbers.
i.e. 0, 1, 2, 3, ... are whole numbers.
To check whether a number is divisible by another number
A given number is said to be divisible by another number if there is no remainder when the given
 ‘No remainder’ has the same
meaning as ‘the remainder is 0’.
number is divided by the other number.
e.g.
5
4 21
20
1
5
4 20
20
20 ÷ 4 = 5
21 ÷ 4 = 5 ... 1
So, 20 is divisible by 4 while 21 is not divisible by 4.
Review of the tests of divisibility of 2, 5 and 10

Divisibility by 2
A whole number is divisible by 2 when its unit digit is 0, 2, 4, 6 or 8.
e.g. 34, 62, 256 and 8710 are divisible by 2. But 51, 73 and 185 are not divisible by 2.

Divisibility by 10
A whole number is divisible by 10 when its unit digit is 0.
e.g. 40, 110, 560 and 9130 are divisible by 10. But 32, 164 and 747 are not divisible by 10.

Divisibility by 5
A whole number is divisible by 5 when its unit digit is 5 or 0.
e.g. 35, 190, 685 and 4590 are divisible by 5. But 46, 218 and 549 are not divisible by 5.
Junior Secondary Mathematics in Action
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The Test of Divisibility of 3
Classwork
1.
2.
3.
Determine whether each of the following numbers is divisible by 2.
(a) 147
_______________
(b) 3408
_______________
(c) 16536
_______________
Determine whether each of the following numbers is divisible by 5.
(a) 265
_______________
(b) 6510
_______________
(c) 41949
_______________
Determine whether each of the following numbers is divisible by 10.
(a) 720
_______________
(b) 1375
_______________
(c) 27580
_______________
It is easy to determine whether a number is divisible by 2, 5 or 10 by checking its unit digit.
Now, let us explore a test to check whether a number is divisible by 3.
The following numbers are all divisible by 3.
12 ÷ 3 = 4
15 ÷ 3 = 5
18 ÷ 3 = 6
21 ÷ 3 = 7
24 ÷ 3 = 8
27 ÷ 3 = 9
30 ÷ 3 = 10
33 ÷ 3 = 11
36 ÷ 3 = 12
39 ÷ 3 = 13
42 ÷ 3 = 14
45 ÷ 3 = 15
12, 15, 18, 21, ... are divisible by 3. Their unit digits include every
number from 0 to 9, and there is no clear pattern for their unit digits.
Junior Secondary Mathematics in Action
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The Test of Divisibility of 3
Let’s consider all the digits of the numbers.
tens
digit
units
digit
Sum of digits
1 2
1+2=3
1 5
1+5=6
1 8
1+8=9
2 1
2+1=3
2 4
2+4=6
2 7
2+7=9
3 0
3+0=3
3 3
3+3=6
3 6
3+6=9
3 9
3 + 9 = 12
4 2
4+2=6
4 5
4+5=9
We can notice that the sum of the digits of these numbers are all divisible by 3.
Is it true for all other numbers? Let’s check through the following activity.
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The Test of Divisibility of 3
Activity
1.
The following numbers are all divisible by 3.
Number
Sum of digits
63
____ + ____ = ____
96
____ + ____ = ____
471
____ + ____ + ____ = ____
855
____ + ____ + ____ = ____
(a) Find the sum of digits of these numbers by completing the table.
(b) Are all the sums of digits divisible by 3?
 Yes
2.
 No
The following numbers are NOT divisible by 3.
Number
Sum of digits
56
____ + ____ = ____
89
____ + ____ = ____
253
____ + ____ + ____ = ____
374
____ + ____ + ____ = ____
(a) Find the sum of digits of these numbers by completing the table.
(b) Are all the sums of digits divisible by 3?
 Yes
 No
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The Test of Divisibility of 3
3.
Now, try to input some 3-digit numbers and complete the following table.
(Part (a) has been done for you as an example.)
3-digit number
(a)
165
Divide the number
by 3
55
3 165
15
15
15
Sum of digits
Is the sum of digits
divisible by 3?
1 + 6 + 5 = 12
 Yes

 No
(b)
 Yes
 No
(c)
 Yes
 No
(d)
 Yes
 No
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The Test of Divisibility of 3
In fact, we have the following test of divisibility of 3.

If the sum of digits of a whole number is divisible by 3,
then the number is divisible by 3.

If the sum of digits of a whole number is not divisible by 3,
then the number is not divisible by 3.
Classwork
1.
Consider the following numbers.
82, 105, 227, 348, 499, 771, 1042, 1869
Which of them are divisible by 3?
2.
_____________________________
Consider the following numbers.
74, 162, 290, 357, 684, 905, 1341, 1673
Which of them are not divisible by 3?
_____________________________
Summary
If the sum of the digits of a whole number is divisible by 3,
then the number is divisible by 3.
For example:
(a) For the number 57,
5 + 7 = 12 and 12 is divisible by 3.
So, 57 is divisible by 3.
(b) For the number 873,
8 + 7 + 3 = 18 and 18 is divisible by 3.
So, 873 is divisible by 3.
(c) For the number 361,
3 + 6 + 1 = 10 and 10 is not divisible by 3.
So, 361 is not divisible by 3.
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The Test of Divisibility of 3
Exercise
1.
Circle all the numbers which are divisible by 3.
72
2.
146
330
528
691
714
1152
2021
Write down the largest 3-digit number which is divisible by 3.
__________________________________________________________________
3.
Write down the smallest 6-digit number which is divisible by 3.
__________________________________________________________________
4.
Write down three 4-digit numbers which are divisible by 3.
__________________________________________________________________
5.
Write down three 5-digit numbers which are divisible by 3.
__________________________________________________________________
6.
The 3-digit number 57 is divisible by 3. Which digit(s) may  represent?
__________________________________________________________________
7.
Let  represent one of the digits 0, 1, 2, ..., 9. If 4,  and 9 is arranged to form a
3-digit number which is divisible by 3, what is the smallest possible value of ?
__________________________________________________________________
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