The LCM of two numbers is 14 times their HCF. The sum of their HCF and LCM is 600. If one number is 280, then find the other number.

Answer

To solve the problem step by step, we will follow the relationships given in the question.

### Step 1: Define the variables
Let:
- HCF = \( h \)
- LCM = \( l \)
- First number = 280
- Second number = \( x \)

### Step 2: Use the relationships given in the problem
According to the problem:
1. \( l = 14h \) (The LCM is 14 times the HCF)
2. \( h + l = 600 \) (The sum of HCF and LCM is 600)

### Step 3: Substitute the LCM in the sum equation
From the first relationship, we can substitute \( l \) in the second equation:
\[ h + 14h = 600 \]
This simplifies to:
\[ 15h = 600 \]

### Step 4: Solve for HCF
Now, divide both sides by 15 to find \( h \):
\[ h = \frac{600}{15} = 40 \]

### Step 5: Find the LCM
Now that we have \( h \), we can find \( l \):
\[ l = 14h = 14 \times 40 = 560 \]

### Step 6: Use the relationship between LCM, HCF, and the two numbers
We know that:
\[ \text{LCM} \times \text{HCF} = \text{First number} \times \text{Second number} \]
Substituting the values we have:
\[ 560 \times 40 = 280 \times x \]

### Step 7: Solve for the second number \( x \)
Calculating the left side:
\[ 22400 = 280 \times x \]
Now, divide both sides by 280:
\[ x = \frac{22400}{280} = 80 \]

### Final Answer
The other number is \( 80 \).

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