Answer :
Sure! Let's find the highest common factor (HCF) and lowest common multiple (LCM) of the numbers 72, 90, and 126 step-by-step.
### 2.3.1 The Highest Common Factor (HCF)
The highest common factor (HCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides each of the given numbers without leaving a remainder.
Step 1: Find the prime factorization of each number.
- [tex]\(72 = 2^3 \times 3^2\)[/tex]
- [tex]\(90 = 2 \times 3^2 \times 5\)[/tex]
- [tex]\(126 = 2 \times 3^2 \times 7\)[/tex]
Step 2: Identify the common prime factors.
- All three numbers have the prime factors [tex]\(2\)[/tex] and [tex]\(3^2\)[/tex].
Step 3: Determine the smallest power of each common prime factor in all numbers.
- The common prime factor is 3, and the smallest power is [tex]\(3^2\)[/tex].
Step 4: Calculate the HCF.
- Therefore, HCF = [tex]\(3^2 = 9\)[/tex].
### 2.3.2 The Lowest Common Multiple (LCM)
The lowest common multiple (LCM) is the smallest positive integer that is divisible by each of the given numbers.
Step 1: Again, use the prime factorization.
- [tex]\(72 = 2^3 \times 3^2\)[/tex]
- [tex]\(90 = 2 \times 3^2 \times 5\)[/tex]
- [tex]\(126 = 2 \times 3^2 \times 7\)[/tex]
Step 2: List all prime factors with their highest powers across all numbers.
- The prime factor 2 appears as [tex]\(2^3\)[/tex].
- The prime factor 3 appears as [tex]\(3^2\)[/tex].
- The prime factor 5 appears as [tex]\(5^1\)[/tex].
- The prime factor 7 appears as [tex]\(7^1\)[/tex].
Step 3: Multiply these together to get the LCM.
- LCM = [tex]\(2^3 \times 3^2 \times 5 \times 7\)[/tex]
- LCM = [tex]\(8 \times 9 \times 5 \times 7 = 2520\)[/tex].
Therefore, the HCF of 72, 90, and 126 is 18, and the LCM is 2520.
### 2.3.1 The Highest Common Factor (HCF)
The highest common factor (HCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides each of the given numbers without leaving a remainder.
Step 1: Find the prime factorization of each number.
- [tex]\(72 = 2^3 \times 3^2\)[/tex]
- [tex]\(90 = 2 \times 3^2 \times 5\)[/tex]
- [tex]\(126 = 2 \times 3^2 \times 7\)[/tex]
Step 2: Identify the common prime factors.
- All three numbers have the prime factors [tex]\(2\)[/tex] and [tex]\(3^2\)[/tex].
Step 3: Determine the smallest power of each common prime factor in all numbers.
- The common prime factor is 3, and the smallest power is [tex]\(3^2\)[/tex].
Step 4: Calculate the HCF.
- Therefore, HCF = [tex]\(3^2 = 9\)[/tex].
### 2.3.2 The Lowest Common Multiple (LCM)
The lowest common multiple (LCM) is the smallest positive integer that is divisible by each of the given numbers.
Step 1: Again, use the prime factorization.
- [tex]\(72 = 2^3 \times 3^2\)[/tex]
- [tex]\(90 = 2 \times 3^2 \times 5\)[/tex]
- [tex]\(126 = 2 \times 3^2 \times 7\)[/tex]
Step 2: List all prime factors with their highest powers across all numbers.
- The prime factor 2 appears as [tex]\(2^3\)[/tex].
- The prime factor 3 appears as [tex]\(3^2\)[/tex].
- The prime factor 5 appears as [tex]\(5^1\)[/tex].
- The prime factor 7 appears as [tex]\(7^1\)[/tex].
Step 3: Multiply these together to get the LCM.
- LCM = [tex]\(2^3 \times 3^2 \times 5 \times 7\)[/tex]
- LCM = [tex]\(8 \times 9 \times 5 \times 7 = 2520\)[/tex].
Therefore, the HCF of 72, 90, and 126 is 18, and the LCM is 2520.
