Find the greatest common factor of 84 and 36, showing how you got your answer. Then, using complete sentences, explain what your answer means.
The Correct Answer and Explanation is:
To find the greatest common factor (GCF) of 84 and 36, we first need to find the prime factorizations of each number:
- Prime factorization of 84:
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
84=22×3×784 = 2^2 \times 3 \times 784=22×3×7. - Prime factorization of 36:
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
36=22×3236 = 2^2 \times 3^236=22×32.
Now, to find the GCF, we identify the common factors between the two prime factorizations:
- Both 84 and 36 have 222^222 and 333 as common factors.
The greatest common factor is found by multiplying these common prime factors:
- 22×3=4×3=122^2 \times 3 = 4 \times 3 = 1222×3=4×3=12.
Explanation:
The greatest common factor (GCF) of 84 and 36 is 12. This means that 12 is the largest number that divides both 84 and 36 without leaving a remainder. In other words, 12 is the biggest number that both 84 and 36 share in their prime factorizations. If you were to divide both 84 and 36 by 12, the result would be whole numbers:
- 84 ÷ 12 = 7
- 36 ÷ 12 = 3
The concept of the GCF is useful in simplifying fractions, finding common denominators, and solving problems in number theory. In practical terms, knowing the GCF helps when we need to break down numbers into smaller parts that still share a common factor.
