Find the HCF of 150 and 225 using the prime factorization method. Answer in full definition.
The correct answer and explanation is:
Answer:
The Highest Common Factor (HCF) of 150 and 225 is 75.
Full Explanation Using Prime Factorization Method:
Definition:
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more numbers is the largest number that divides all of them exactly without leaving a remainder.
Step 1: Prime Factorization of Each Number
- Prime factorization is the process of expressing a number as the product of its prime factors.
- Find the prime factors of 150:
150 ÷ 2 = 75
75 ÷ 3 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1 So, prime factors of 150 are: 150=2×3×5×5=2×3×52150 = 2 \times 3 \times 5 \times 5 = 2 \times 3 \times 5^2 - Find the prime factors of 225:
225 ÷ 3 = 75
75 ÷ 3 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1 So, prime factors of 225 are: 225=3×3×5×5=32×52225 = 3 \times 3 \times 5 \times 5 = 3^2 \times 5^2
Step 2: Identify the Common Prime Factors
- Look at the prime factors of both numbers:
- 150 = 2×3×522 \times 3 \times 5^2
- 225 = 32×523^2 \times 5^2
- Common prime factors are those present in both numbers with the lowest powers:
- For 3: minimum power is 313^1 (because 150 has one 3, and 225 has two 3s)
- For 5: minimum power is 525^2 (both have two 5s)
- For 2: 225 does not have 2, so exclude it.
Step 3: Multiply the common factors HCF=31×52=3×25=75HCF = 3^1 \times 5^2 = 3 \times 25 = 75
Summary:
- The HCF is found by prime factorizing both numbers.
- Common prime factors with the lowest exponent are multiplied.
- For 150 and 225, the HCF is 75.
Why is HCF important?
- HCF is useful in simplifying fractions, finding common denominators, and solving problems involving divisibility.
- It represents the greatest number that can evenly divide the given numbers, showing their highest shared factor.