GCF of 36 and 48
GCF of 36 and 48
The Greatest Common Factor (GCF) of 36 and 48 is determined by finding the largest factor that both numbers share. Using the prime factorization method, 36 is factored into 2² × 3², and 48 is factored into 2⁴ × 3. The common prime factors are 2 and 3, with the lowest powers being 2² and 3¹. Multiplying these common prime factors, we get 2² × 3 = 4 × 3 = 12. Therefore, the GCF of 36 and 48 is 12. This can also be confirmed by listing the factors of each number and identifying the largest common factor.
GCF of 36 and 48
GCF of 36 and 48 is 12.
Methods to Find GCF of 36 and 48
- Prime Factorization Method
- Long Division Method
- Listing Common Factors
GCF of 36 and 48 by Prime Factorization Method.
To find the Greatest Common Factor (GCF) of 36 and 48 using the prime factorization method, follow these steps:
Step-by-Step Process:
Prime Factorization of Each Number:
Prime factors of 36:
36 = 2 × 2 × 3 × 3
36 = 2² × 3²
Prime factors of 48:
48 = 2 × 2 × 2 × 2 × 3
48 = 2⁴ × 3
Identify the Common Prime Factors:
The common prime factors are 2 and 3.
The lowest power of 2 common to both numbers is 2².
The lowest power of 3 common to both numbers is 3¹.
Multiply the Common Prime Factors:
GCF = 2² × 3¹
GCF = 4 × 3
GCF = 12
GCF of 36 and 48 by Long Division Method.
To find the Greatest Common Factor (GCF) of 36 and 48 using the Long Division Method, follow these steps:
Step-by-Step Process:
Divide the Larger Number by the Smaller Number:
Divide 48 (larger number) by 36 (smaller number).
48 ÷ 36 = 1 remainder 12
Replace the Larger Number with the Smaller Number:
The divisor (36) becomes the new dividend.
The remainder (12) becomes the new divisor.
Repeat the Division:
Now, divide 36 by 12.
36 ÷ 12 = 3 remainder 0
Check the Remainder:
When the remainder is 0, the current divisors is the GCF.
The remainder is 0, and the current divisor is 12.
GCF of 36 and 48 by Listing Common Factors.
To find the Greatest Common Factor (GCF) of 36 and 48 by listing their common factors, follow these steps:
Step-by-Step Process:
List the Factors of Each Number:
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Identify the Common Factors:
The common factors of 36 and 48 are: 1, 2, 3, 4, 6, 12
Find the Greatest Common Factor:
The largest number in the list of common factors is 12.
Can the GCF of 36 and 48 be found using the Euclidean algorithm?
Yes, the Euclidean algorithm involves repeated division. Divide 48 by 36 to get a remainder of 12. Then, divide 36 by 12 to get a remainder of 0. The GCF is 12.
What is the relationship between the GCF and LCM of 36 and 48?
The product of the GCF and LCM of two numbers equals the product of the numbers. For 36 and 48: GCF × LCM = 36 × 48.
How can finding the GCF of 36 and 48 help in real-life problems?
It helps in tasks such as simplifying recipes, splitting objects into equal parts, and solving problems involving proportions or ratios.
How does the GCF help in simplifying fractions involving 36 and 48?
The GCF helps reduce fractions to their simplest form. For example, 36/48 simplifies to 3/4 when both numerator and denominator are divided by their GCF, which is 12.
How is the GCF used in solving ratio problems?
The GCF is used to simplify ratios. For example, the ratio 36:48 simplifies to 3:4 by dividing both terms by their GCF, which is 12.
Is the GCF of two numbers always smaller than the numbers themselves?
Yes, the GCF is always less than or equal to the smaller of the two numbers.
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