The highest common factor (HCF) of 12, 15 and 60 is
The correct answer and explanation is:
Answer: The highest common factor (HCF) of 12, 15, and 60 is 3.
Explanation:
The Highest Common Factor (HCF), also called the Greatest Common Divisor (GCD), of a set of numbers is the largest number that divides all of them exactly without leaving a remainder.
To find the HCF of 12, 15, and 60, we can follow these steps:
1. Prime Factorization
First, express each number as a product of its prime factors:
- 12 = 2 × 2 × 3 = 22×32^2 \times 3
- 15 = 3 × 5 = 3×53 \times 5
- 60 = 2 × 2 × 3 × 5 = 22×3×52^2 \times 3 \times 5
2. Identify Common Prime Factors
Now, look at the prime factors common to all three numbers:
- For 12, the prime factors are 2 and 3.
- For 15, the prime factors are 3 and 5.
- For 60, the prime factors are 2, 3, and 5.
The only prime factor common to all three numbers is 3.
3. Determine the Lowest Power of Common Factors
Since 3 appears once in each number’s prime factorization, the lowest power of 3 common to all is 31=33^1 = 3.
4. Calculate the HCF
The HCF is the product of the common prime factors raised to their lowest powers: HCF=3\text{HCF} = 3
Alternative Method: Listing Factors
You can also find the HCF by listing the factors of each number:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 15: 1, 3, 5, 15
- Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
The common factors are 1 and 3. The greatest of these is 3.
Why is HCF important?
The HCF is useful in simplifying fractions, solving problems involving divisibility, and working with ratios. It helps to reduce numbers to their simplest form by dividing both the numerator and denominator by their HCF.
In this case, the HCF of 12, 15, and 60 being 3 means 3 is the largest number that can divide all three numbers without leaving a remainder.