What is the highest common factor (HCF) of 8, 16 and 18?
The Correct Answer and Explanation is:
The highest common factor (HCF), also known as the greatest common divisor (GCD), of 8, 16, and 18 is 2.
Here’s how to find the HCF of 8, 16, and 18:
Step 1: Prime Factorization
We start by finding the prime factorization of each number.
- 8: The prime factorization of 8 is 8=2×2×2=238 = 2 \times 2 \times 2 = 2^38=2×2×2=23.
- 16: The prime factorization of 16 is 16=2×2×2×2=2416 = 2 \times 2 \times 2 \times 2 = 2^416=2×2×2×2=24.
- 18: The prime factorization of 18 is 18=2×3×3=2×3218 = 2 \times 3 \times 3 = 2 \times 3^218=2×3×3=2×32.
Step 2: Identify Common Prime Factors
Now, we look at the prime factors of each number:
- 8 has 232^323.
- 16 has 242^424.
- 18 has 21×322^1 \times 3^221×32.
The only common prime factor among all three numbers is 2. The smallest power of 2 that appears in the factorization of all three numbers is 212^121, which equals 2.
Step 3: Conclusion
The highest common factor of 8, 16, and 18 is 2 because it is the largest number that divides all three without leaving a remainder.
Why is this important?
The HCF is useful for simplifying fractions, determining common denominators in fractions, and solving problems in number theory. For example, if you wanted to divide these numbers into equal groups or portions, 2 would be the largest number you could use to ensure each group had an equal size without leftovers.
So, the HCF of 8, 16, and 18 is 2.
