How do you find the common factors of two whole numbers?
I. Factors and multiples
A. Definitions
a and b are two whole numbers.
b is a factor of a if a whole number, q, exists, such that a = b × q.
a is also referred to as a multiple of b, i.e., a is divisible by b.
Notes:
If b is a factor of a, then the division of a by b gives a remainder of 0. Thus, we can write a = b × q, where q is the quotient of a divided by b;
Some calculators have a key for division where sometimes both the quotient and remainder are displayed on the screen.
Example: Are 13 and 7 factors of 221?
Carry out division of 221 by 13, then 221 by 7:
221 = 13 × 17, therefore 13 is a factor of 221.
221 = (7 × 31) + 4, therefore 7 is not a factor of 221.
B. Reminder of tests of divisibility
It is not always necessary to perform a division to know if a whole number is divisible by another; the following rules serve as a reminder:
A whole number is divisible by 2 if its last digit is zero, 2, 4, 6, or 8.
A whole number is divisible by 3 if the sum of its digits is divisible by 3.
A whole number is divisible by 5 if its last digit is zero or 5.
A whole number is divisible by 9 if the sum of its figures is divisible by 9.
A whole number is divisible by 10 if its last digit is zero.
Example: According to these criteria, we can say that 975 is divisible by 3 and 5, but is not divisible by 2, 9, or 10.
II. The factors of a whole number
A. Some rules
Consider a whole number a other than zero or 1. This number has at least two factors: 1 and itself.
It is always true that a = a × 1.
The number 1 only has one factor: itself.
The number zero is divisible by every non-zero whole number.
Often, the factors of a whole number can be paired: For example, 8 and 9 are a pair of factors of 72, because 72 is divisible by 8 and 9, and 72 = 8 × 9.
B. Method
The method of finding all the factors of a whole number can be explained using an example: Find all the factors of 72. This can be done by dividing 72 by every successive whole number: 1, 2, 3, etc.
Where the remainder is zero, write the corresponding equation and the pair of factors obtained.
The process stops because the following equation, 72 = 9 × 8, gives the factors 8 and 9, which have already been obtained.
The factors of 72 are therefore: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
III. The common factors of two whole numbers
A. Example
Find all the common factors of 72 and 54. To do this, find the factors of each of these numbers using the method explained above, then take the numbers that appear in both lists:
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, and 54.
The common factors of 72 and 54 are therefore: 1, 2, 3, 6, 9, and 18.
B. Definition of the HCF
The highest common factor (or greatest common divisor) of two whole numbers is, in abbreviated form, the HCF of these two whole numbers.
Example: The HCF of 72 and 54 is 18 (from the example given above).
This is written: HCF(72, 54) = 18.
Returning to the list of common factors of 72 and 54: 1, 2, 3, 6, 9, and 18, note that these are all factors of 18.
Property: The common factors of two whole numbers are the factors of their HCF. If we know the HCF of two whole numbers, we can just find all of its factors to find the common factors of these two whole numbers.
Read more:
Applying the Distributive Law
Calculating a Numerical Expression (1)
Calculating a Numerical Expression (2)
Recognizing a Proportional Relationship
Rounding to the Nearest Unit
Writing a Numerical Expression That Corresponds to a Sequence of Operations
Read more:
Applying the Distributive Law
Calculating a Numerical Expression (1)
Calculating a Numerical Expression (2)
Recognizing a Proportional Relationship
Rounding to the Nearest Unit
Writing a Numerical Expression That Corresponds to a Sequence of Operations