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Skillsheet
Factors and divisibility
A factor (or divisor) of a number is a whole number that divides into that number exactly. For example, 2 is
one factor of 18 because 2 divides into 18 (18 2 ¼ 9). The number 6 is another factor of 18 because 6 divides
into 18 (18 6 ¼ 3). However, 4 is not a factor of 18 because 4 does not divide into 18 exactly.
Divisibility tests
To check whether a number is a factor of a given number, we can use the tests for divisibility.
Divisibility by
Test
2
if the number ends in 2, 4, 6, 8 or 0
3
if the sum of the number’s digits is divisible by 3
5
if the number ends in 5 or 0
6
if the number is divisible by both 2 and 3
9
if the sum of the number’s digits is divisible by 9
10
if the number ends in 0
Example 1
Test whether each of these numbers is a factor of 50.
a 2
b 3
c 6
d 10
Solution
a The test for divisibility by 2 is that the number ends in 2, 4, 6, 8 or 0. The number 50 ends in 0, so it is
divisible by 2, so 2 is a factor of 50.
b The test for divisibility by 3 is to add the digits of the number and see if the sum of the digits is divisible
by 3. Using the digits of 50, we see that 5 þ 0 ¼ 5, which is not divisible by 3. So 50 is not divisible by 3,
and 3 is not a factor of 50.
c The test for divisibility by 6 is that the number is divisible by both 2 and 3. From the above working, 50 is
divisible by 2 but not divisible by 3. So 50 is not divisible by 6, and 6 is not a factor of 50.
d The test for divisibility by 10 is that the number ends in 0. The number 50 ends in 0, so it is divisible by
10 and 10 is a factor of 50.
© Cengage Learning Australia Pty Ltd 2011
MAT07NASS10014
Number and Algebra: Number and place value
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Example 2
Test whether:
a 9 is a factor of 49
b 5 is a factor of 80
c 3 is a factor of 29
d 6 is a factor of 96
Solution
a The test for divisibility by 9 is to add the digits and see if the sum is divisible by 9. Using the digits of 49,
we see that 4 þ 9 ¼ 13, which is not divisible by 9. So 9 is not a factor of 49.
b The test for divisibility by 5 is that the number ends in 5 or 0. The number 80 ends in 0, so it is divisible by
5, and 5 is a factor of 80.
c The test for divisibility by 3 is that the sum of the digits is also divisible by 3. For 29 we see that 2 þ 9 ¼
11, which is not divisible by 3, so 3 is not a factor of 29.
d The test for divisibility by 6 is that the number is divisible by both 2 and 3. The number 96 is divisible by 2
because it ends in 6. Testing for divisibility by 3, we see that 9 þ 6 ¼ 15, which is divisible by 3. Therefore,
96 is divisible by 6, and 6 is a factor of 96.
Exercise
1 Test whether:
a 3 is a factor of 54
b 2 is a factor of 44
c 10 is a factor of 200
d 9 is a factor of 84
e 3 is a factor of 47
f 5 is a factor of 90
g 6 is a factor of 72
h 9 is a factor of 108
i 6 is a factor of 99
Common factors
Example 3
Find a common factor for each pair of numbers.
a 27 and 45
b 40 and 75
c 24 and 36
Solution
a
27 and 45 are both not divisible by 2 so 2 is not a common factor.
Try divisibility by 3.
2 þ 7 ¼ 9, which is divisible by 3, and 4 þ 5 ¼ 9, which is divisible by 3.
So 3 is a common factor of 27 and 45. (Another possible answer is 9).
b
40 is divisible by 2 but 75 is not, so 2 is not a common factor.
However, 40 and 75 both end in 5 or 0, so both are divisible by 5.
So 5 is a common factor of 40 and 75.
c
24 and 36 are both divisible by 2, so 2 is a common factor.
(Other possible answers are 3, 4, 6 and 12).
© Cengage Learning Australia Pty Ltd 2011
MAT07NASS10014
Number and Algebra: Number and place value
www.nelsonnet.com.au
[Satish][NelsonNet Worksheet][D:/Thomson_Learning_Projects/NelsonNet_Worksheets_SB7_1080060/z_production/z_3B2_3D_files/Maths worksheet MS x 43/Student resources/NCM_Batch1/Skillsheets/MAT07NASS10014_Factors_and_divisibility.3d] [26/1/012/0:41:3] [1–4]
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Exercise
2 Find a common factor for each pair of numbers.
a 32 and 50
b 20 and 80
c 33 and 72
d 45 and 75
e 44 and 56
f 27 and 69
g 45 and 90
h 48 and 60
i 46 and 64
j 80 and 120
k 85 and 115
l 78 and 102
© Cengage Learning Australia Pty Ltd 2011
MAT07NASS10014
Number and Algebra: Number and place value
www.nelsonnet.com.au
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Answers
1 a Yes
b Yes
c Yes
d No
f Yes
g Yes
h Yes
b 2, 4, 5, 10 or 20
c 3
d 3, 5 or 15
e 2 or 4
f 3
g 3, 5, 9, 15 or 45
h 2, 3, 4, 6 or 12
i 2
j 2, 4, 5, 8, 10, 20 or 40
k 5
e No
i No
2 a 2
© Cengage Learning Australia Pty Ltd 2011
MAT07NASS10014
l 2, 3 or 6
Number and Algebra: Number and place value
www.nelsonnet.com.au