what is the greatest common factor of 27, 45, 63
The Correct Answer and Explanation is:
Correct Answer: 9
To find the greatest common factor (GCF) of the numbers 27, 45, and 63, we need to determine the largest number that divides all three without leaving a remainder. This process involves finding the prime factorizations of each number.
Step 1: Prime factorization
- 27 = 3 × 3 × 3 = 3³
- 45 = 3 × 3 × 5 = 3² × 5
- 63 = 3 × 3 × 7 = 3² × 7
Step 2: Identify the common prime factors
Look at each factorization and identify the prime numbers that appear in all three. The only common prime factor is 3. We then choose the smallest exponent of that prime factor among the numbers.
- 27 has 3³
- 45 has 3²
- 63 has 3²
The smallest power of 3 that appears in all three numbers is 3², which equals 9.
Step 3: Conclusion
So, the greatest common factor is 9.
Explanation
The greatest common factor (GCF) is the largest number that evenly divides two or more numbers. To find the GCF of 27, 45, and 63, it is helpful to use prime factorization. This method breaks down each number into its basic building blocks, or prime numbers.
We start with 27. Since 27 is divisible by 3, we divide repeatedly:
27 ÷ 3 = 9,
9 ÷ 3 = 3,
3 ÷ 3 = 1.
So, 27 = 3³.
Next is 45. Divide by 3 to get 15, then divide 15 by 3 to get 5. So,
45 = 3² × 5.
Now for 63. Divide 63 by 3 to get 21, then divide 21 by 3 to get 7. So,
63 = 3² × 7.
After factoring all three numbers, we look for common prime factors. All three have the prime number 3 in common. The smallest power of 3 that appears in each is 3², or 9. This means 9 is the highest number that divides evenly into all three numbers.
Understanding how to find the GCF is useful in many areas of math, especially when simplifying fractions or solving problems that involve ratios. It helps ensure that numbers are reduced to their simplest form and calculations are accurate.
