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

Answer

To solve the problem step by step, we will follow the information given in the question and apply the relevant mathematical concepts.

### Step 1: Define Variables
Let:
- LCM = M
- HCF = N

### Step 2: Set Up Equations
From the problem, we have two key pieces of information:
1. The LCM of two numbers is 14 times their HCF:
\[ M = 14N \]

2. The sum of LCM and HCF is 600:
\[ M + N = 600 \]

### Step 3: Substitute for HCF
From the second equation, we can express N in terms of M:
\[ N = 600 - M \]

### Step 4: Substitute N into the LCM Equation
Now substitute \(N\) from the above equation into the first equation:
\[ M = 14(600 - M) \]

### Step 5: Expand and Rearrange
Expanding the equation gives:
\[ M = 8400 - 14M \]
Now, rearranging this equation to isolate M:
\[ M + 14M = 8400 \]
\[ 15M = 8400 \]

### Step 6: Solve for LCM
Now, divide both sides by 15 to find M:
\[ M = \frac{8400}{15} = 560 \]

### Step 7: Find HCF
Now that we have the LCM, we can find the HCF using:
\[ N = 600 - M = 600 - 560 = 40 \]

### Step 8: Use the Relationship Between LCM, HCF, and the Numbers
We know that:
\[ \text{LCM} \times \text{HCF} = \text{Product of the two numbers} \]
Substituting the values we have:
\[ 560 \times 40 = 280 \times x \]
Where \(x\) is the other number we need to find.

### Step 9: Calculate the Product
Calculating the left side:
\[ 560 \times 40 = 22400 \]
So we have:
\[ 22400 = 280 \times x \]

### Step 10: Solve for the Other Number
Now, divide both sides by 280 to find \(x\):
\[ x = \frac{22400}{280} = 80 \]

### Final Answer
Thus, the other number is:
\[ \boxed{80} \]