Finding Factors
Examples with Solutions
EXAMPLE 1
What are the factors of 45?
Solution
Let’s see if 45 is divisible by 1, 2, 3, and so on, using
the divisibility tests wherever they apply.
Is 45 divisible by

Answer

1? 
Yes, because 1 is a factor of any number;
, so 45 is also a factor. 
2? 
No, because the ones digit is not even. 
3? 
Yes, because the sum of the digits, 4 + 5
= 9, is divisible by 3; , so 15 is also a factor. 
4? 
No, because 4 will not divide into 45
evenly. 
5? 
Yes, because the ones digit is 5; , so 9 is also a factor. 
6? 
No, because 45 is not even. 
7? 
No, because 45 Ã· 7 has remainder 3. 
8? 
No, because 45 Ã· 8 has remainder 5. 
9? 
We already know that 9 is a factor. 
The factors of 45 are therefore 1, 3, 5, 9, 15, and 45.
Note that we really didn’t have to check to see if 9 was
a factor—we learned that itwas when we checked for
divisibility by 5. Also, because the factors were beginning
torepeat with 9, there was no need to check numbers greater than
9.
EXAMPLE 2
Identify all the factors of 60.
Solution
Let’s check to see if 60 is divisible by 1, 2, 3, 4, and
so on.
Is 60 divisible by

Answer

1? 
Yes, because 1 is a factor of all
numbers; , so 60 is also a factor. 
2? 
Yes, because the ones digit is even; , so 30 is also a factor. 
3? 
Yes, because the sum of the digits, 6 + 0
= 6, is divisible by 3; , so 20 is also a factor. 
4? 
Yes, because 4 will divide into 60
evenly; , so 15 is also a factor. 
5? 
Yes, because the ones digit is 0; , so 12 is also a factor. 
6? 
Yes, because the ones digit is even and
the sum of the digits is divisibleby 3; , so 10 is also a factor. 
7? 
No, because 60 Ã· 7 has remainder 4. 
8? 
No, because 60 Ã· 8 has remainder 4. 
9? 
No, because the sum of the digits, 6 + 0
= 6, is not divisible by 9. 
10? 
We already know that 10 is a factor. 
The factors of 60 are therefore 1, 2, 3, 4, 5, 6, 10, 12, 15,
20, 30, and 60. Can you explain how we knew that 10 was a factor
of 60 when we checked for divisibility by 6?
EXAMPLE 3
A presidential election takes place in the United States every
year thatis a multiple of 4. Was there a presidential election in
1866?
Solution
The question is: Does 4 divide into 1866 evenly? Using the
divisibility test for 4, we check whether 66 is a multiple of 4.
Because has remainder 2, 4 is not a factor of 1866. So there
was nopresidential election in 1866.
