Find the LCM and HCF of the of the following integers by applying the prime factorisation method.
8, 9 and 25

Answer

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the integers 8, 9, and 25 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **For 8**:
- 8 can be divided by 2:
- \( 8 \div 2 = 4 \)
- \( 4 \div 2 = 2 \)
- \( 2 \div 2 = 1 \)
- Therefore, the prime factorization of 8 is \( 2^3 \).

- **For 9**:
- 9 can be divided by 3:
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- Therefore, the prime factorization of 9 is \( 3^2 \).

- **For 25**:
- 25 can be divided by 5:
- \( 25 \div 5 = 5 \)
- \( 5 \div 5 = 1 \)
- Therefore, the prime factorization of 25 is \( 5^2 \).

### Step 2: Write Down the Prime Factorizations
Now we can summarize the prime factorizations:
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)
- \( 25 = 5^2 \)

### Step 3: Calculate HCF
To find the HCF, we look for the common prime factors with the lowest powers:
- The prime factors of 8, 9, and 25 are 2, 3, and 5 respectively.
- There are no common prime factors among all three numbers.

Thus, the HCF is:
\[ \text{HCF} = 1 \]

### Step 4: Calculate LCM
To find the LCM, we take the highest power of each prime factor present in any of the numbers:
- From 8, we take \( 2^3 \)
- From 9, we take \( 3^2 \)
- From 25, we take \( 5^2 \)

Now we multiply these together:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^2 \]

Calculating this:
- \( 2^3 = 8 \)
- \( 3^2 = 9 \)
- \( 5^2 = 25 \)

Now multiply:
\[ \text{LCM} = 8 \times 9 \times 25 \]
Calculating step by step:
- \( 8 \times 9 = 72 \)
- \( 72 \times 25 = 1800 \)

Thus, the LCM is:
\[ \text{LCM} = 1800 \]

### Final Results
- **HCF**: 1
- **LCM**: 1800

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