Greatest Common Factor (GCF/HCF) - Sixth Grade Mathematics
The Greatest Common Factor, also known as the Highest Common Factor, is the largest positive whole number that divides evenly into two or more numbers without leaving a remainder.
Conceptual Foundation of Common Factors
When two or more numbers are analyzed, they frequently share several factors in common. The greatest among these shared divisors is the Greatest Common Factor (GCF). Finding the GCF is the primary mathematical tool used to simplify fractions to their simplest form and to factor algebraic expressions.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Shared Common Factors: { 1, 2, 3, 4, 6, 12 }
Greatest Common Factor: 12
Methods for Calculating GCF
Three proven methods allow you to calculate the GCF of two or more numbers depending on their size: listing factors, prime factorization, and the Euclidean algorithm.
+-------------------------------------------------------------------------+
| THREE METHODS TO FIND GCF |
+-------------------+-----------------------+-----------------------------+
| Method | Procedure | Best Applied To |
+-------------------+-----------------------+-----------------------------+
| Listing Factors | Write all factors of | Small numbers (< 50) with |
| | each number; find max | few total divisors |
+-------------------+-----------------------+-----------------------------+
| Prime | Decompose to primes; | Standard multi-digit |
| Factorization | multiply lowest power | numbers (50 to 500) |
| | of all COMMON primes | |
+-------------------+-----------------------+-----------------------------+
| Euclidean | Successive division | Large multi-digit numbers |
| Algorithm | using remainders | (> 500) |
+-------------------+-----------------------+-----------------------------+
The Prime Factorization Method
To find the GCF of two or more numbers using prime factorization: First, write the prime factorization of each number in exponential form. Second, identify the prime bases that are common to all factorizations. Third, for each common prime base, select the term with the smallest exponent. Fourth, multiply these lowest-power terms together.
Find GCF of 72 and 108:
72 = 2^3 * 3^2
108 = 2^2 * 3^3
Common prime bases: 2 and 3.
Lowest power of 2: 2^2 (from 108)
Lowest power of 3: 3^2 (from 72)
GCF(72, 108) = 2^2 * 3^2 = 4 * 9 = 36.
The Euclidean Algorithm for Large Numbers
When numbers have hundreds or thousands of factors, listing or factoring can become tedious. The Euclidean algorithm uses repeated division. Divide the larger number by the smaller number and note the remainder. Then divide the previous divisor by that remainder. Repeat this cycle until the remainder reaches zero. The last non-zero remainder is the GCF.
Euclidean Algorithm for GCF(252, 105):
Step 1: Divide 252 by 105: 252 = (2 * 105) + 42 (Remainder = 42)
Step 2: Divide 105 by 42: 105 = (2 * 42) + 21 (Remainder = 21)
Step 3: Divide 42 by 21: 42 = (2 * 21) + 0 (Remainder = 0)
Conclusion: The last non-zero remainder is 21. Therefore, GCF(252, 105) = 21.
Finding the GCF of Three Numbers
To find the GCF of three numbers, such as 48, 72, and 120, decompose all three into their prime factorizations: 48 = 2^4 * 3 72 = 2^3 * 3^2 120 = 2^3 * 3 * 5 The common prime bases present in all three numbers are 2 and 3 (the factor 5 is not shared by 48 or 72). The lowest power of 2 is 2^3. The lowest power of 3 is 3^1 = 3. Multiplying gives GCF = 2^3 * 3 = 8 * 3 = 24.
Exercises
Exercise 1. Find the Greatest Common Factor of 54 and 81 using the prime factorization method.
Exercise 2. Use the Euclidean algorithm to find the GCF of 315 and 210.
Exercise 3. Find the GCF of the three numbers 36, 60, and 84.
Solutions
Solution to Exercise 1. Find the prime factorization of each number: 54 = 2 * 27 = 2 * 3^3. For 81: 81 = 9 * 9 = 3^4. The only common prime base shared by both numbers is 3. The lowest exponent of 3 between the two factorizations is 3^3. Therefore, GCF(54, 81) = 3^3 = 27.
Solution to Exercise 2. Apply the Euclidean algorithm to 315 and 210: Divide 315 by 210: 315 = (1 * 210) + 105 (remainder is 105). Next, divide 210 by 105: 210 = (2 * 105) + 0 (remainder is 0). Since the remainder has reached 0, the last non-zero divisor is 105. Therefore, GCF(315, 210) = 105.
Solution to Exercise 3. Decompose all three numbers into primes: 36 = 2^2 * 3^2; 60 = 2^2 * 3 * 5; 84 = 2^2 * 3 * 7. The common prime factors present in all three numbers are 2 and 3. The lowest power of 2 is 2^2. The lowest power of 3 is 3^1. Multiplying these gives GCF = 2^2 * 3 = 4 * 3 = 12.