Find the GCF of 40 and 120.
Answer (2)
The greatest common factor (GCF) of 40 and 120 is 40, found by listing their factors or using prime factorization. Both methods show that 40 is the largest number that divides both without a remainder. Thus, GCF(40, 120) = 40.
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List the factors of 40 and 120.
Identify the common factors of 40 and 120.
Determine the largest common factor, which is 40.
The GCF of 40 and 120 is 40 .
Explanation
Understanding the Problem We need to find the greatest common factor (GCF) of 40 and 120. The GCF is the largest number that divides both 40 and 120 without leaving a remainder.
Listing Factors Let's list the factors of 40: 1, 2, 4, 5, 8, 10, 20, 40. Now, let's list the factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
Identifying Common Factors The common factors of 40 and 120 are: 1, 2, 4, 5, 8, 10, 20, 40.
Determining the GCF The largest number among the common factors is 40. Therefore, the GCF of 40 and 120 is 40.
Prime Factorization Method Alternatively, we can use prime factorization to find the GCF. The prime factorization of 40 is 2 3 × 5 , and the prime factorization of 120 is 2 3 × 3 × 5 . The GCF is found by taking the lowest power of the common prime factors: 2 3 × 5 = 8 × 5 = 40 .
Final Answer Therefore, the greatest common factor of 40 and 120 is 40 .
Examples
Finding the GCF is useful in many real-life situations. For example, suppose you have 40 apples and 120 oranges, and you want to make identical fruit baskets with the same number of apples and oranges in each basket. The GCF of 40 and 120, which is 40, tells you that you can make 40 fruit baskets, each containing 1 apple and 3 oranges. This ensures that you use all the apples and oranges and that each basket is identical.
