To find the two numbers that have an HCF of 15 and a sum of 75, we can follow these steps:
### Step 1: Understand the relationship between the numbers
Let the two numbers be \( a \) and \( b \). We know:
- \( \text{HCF}(a, b) = 15 \)
- \( a + b = 75 \)
### Step 2: Express the numbers in terms of their HCF
Since the HCF of the two numbers is 15, we can express them as:
- \( a = 15m \)
- \( b = 15n \)
where \( m \) and \( n \) are co-prime integers (i.e., their HCF is 1).
### Step 3: Substitute the expressions into the sum equation
Substituting \( a \) and \( b \) into the sum equation gives:
\[ 15m + 15n = 75 \]
### Step 4: Simplify the equation
We can factor out 15 from the left side:
\[ 15(m + n) = 75 \]
Dividing both sides by 15, we get:
\[ m + n = 5 \]
### Step 5: Find pairs of co-prime integers that add up to 5
Now we need to find pairs of integers \( (m, n) \) such that:
- \( m + n = 5 \)
- \( \text{HCF}(m, n) = 1 \)
The possible pairs of integers that satisfy \( m + n = 5 \) are:
1. \( (1, 4) \)
2. \( (2, 3) \)
3. \( (3, 2) \)
4. \( (4, 1) \)
Among these, the pairs \( (1, 4) \) and \( (2, 3) \) are co-prime.
### Step 6: Calculate the corresponding numbers
Now we can calculate the corresponding numbers for each pair:
1. For \( (1, 4) \):
- \( a = 15 \times 1 = 15 \)
- \( b = 15 \times 4 = 60 \)
- The numbers are 15 and 60.
2. For \( (2, 3) \):
- \( a = 15 \times 2 = 30 \)
- \( b = 15 \times 3 = 45 \)
- The numbers are 30 and 45.
### Step 7: Verify the HCF and sum
- For the pair \( (15, 60) \):
- HCF(15, 60) = 15 (correct)
- Sum = 15 + 60 = 75 (correct)
- For the pair \( (30, 45) \):
- HCF(30, 45) = 15 (correct)
- Sum = 30 + 45 = 75 (correct)
### Conclusion
The two pairs of numbers that satisfy the conditions are:
1. 15 and 60
2. 30 and 45
Thus, the numbers are **30 and 45**.
