What is the GCF of 56 and 98?
Understand the Problem
The question is asking for the greatest common factor (GCF) of the numbers 56 and 98, which involves identifying the largest positive integer that divides both numbers without leaving a remainder.
Answer
$14$
Answer for screen readers
The greatest common factor (GCF) of 56 and 98 is $14$.
Steps to Solve
- Finding the prime factorization of 56
To start, we'll find the prime factorization of 56. Divide 56 by the smallest prime numbers:
- $56 \div 2 = 28$
- $28 \div 2 = 14$
- $14 \div 2 = 7$
- $7$ is a prime number.
So, the prime factorization of 56 is: $$ 56 = 2^3 \times 7 $$
- Finding the prime factorization of 98
Next, we'll find the prime factorization of 98 using the same method:
- $98 \div 2 = 49$
- $49 \div 7 = 7$
- $7$ is a prime number.
So, the prime factorization of 98 is: $$ 98 = 2^1 \times 7^2 $$
- Identifying common prime factors
Now, identify the common prime factors from both factorizations:
- For ( 2 ): the lowest power is $2^1$ (from 98)
- For ( 7 ): the lowest power is $7^1$ (from 56)
- Calculating the GCF
Multiply the common prime factors together to find the GCF: $$ \text{GCF} = 2^1 \times 7^1 = 2 \times 7 = 14 $$
The greatest common factor (GCF) of 56 and 98 is $14$.
More Information
The GCF is important in simplifying fractions, finding common denominators, and solving problems involving operations with integers. The GCF method helps in various mathematical concepts, including reducing ratios.
Tips
- Forgetting to factor completely: Ensure that all factors are determined.
- Confusing GCF with LCM: Remember that the GCF is the largest factor that divides both numbers, while the LCM is the smallest multiple that is common.
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