Find the HCF and the LCM for the following sets of numbers
25,35,40

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 25, 35, and 40, we can follow these steps:

### Step 1: Find the Prime Factorization of Each Number
- **25**: The prime factorization of 25 is \(5 \times 5\) or \(5^2\).
- **35**: The prime factorization of 35 is \(7 \times 5\).
- **40**: The prime factorization of 40 is \(2 \times 2 \times 2 \times 5\) or \(2^3 \times 5\).

### Step 2: Identify the Common Factors for HCF
To find the HCF, we look for the common prime factors in all three numbers:
- The only common prime factor among 25, 35, and 40 is \(5\).
- Therefore, the HCF is \(5\).

### Step 3: Identify the Maximum Powers for LCM
To find the LCM, we take the highest power of each prime factor present in any of the numbers:
- For the prime factor \(2\), the highest power is \(2^3\) (from 40).
- For the prime factor \(5\), the highest power is \(5^2\) (from 25).
- For the prime factor \(7\), the highest power is \(7^1\) (from 35).

### Step 4: Calculate the LCM
Now we multiply these together to find the LCM:
\[ \text{LCM} = 2^3 \times 5^2 \times 7^1 \]
Calculating this step by step:
- \(2^3 = 8\)
- \(5^2 = 25\)
- \(7^1 = 7\)

Now multiply these values:
\[ \text{LCM} = 8 \times 25 \times 7 \]
Calculating \(8 \times 25 = 200\), and then \(200 \times 7 = 1400\).

### Conclusion
- **HCF**: \(5\)
- **LCM**: \(1400\)