To solve the problem step by step, we will find two numbers whose HCF is 7, LCM is 140, and that lie between 20 and 45. We will then calculate the sum of those two numbers.
### Step 1: Understand the relationship between HCF, LCM, and the numbers
The relationship between the HCF (Highest Common Factor), LCM (Lowest Common Multiple), and the two numbers (let's call them \(x\) and \(y\)) can be expressed as:
\[
x \times y = \text{HCF} \times \text{LCM}
\]
### Step 2: Substitute the known values
Given that HCF = 7 and LCM = 140, we can substitute these values into the equation:
\[
x \times y = 7 \times 140
\]
Calculating the right side:
\[
x \times y = 980
\]
### Step 3: Express the numbers in terms of HCF
Since the HCF is 7, we can express the two numbers as:
\[
x = 7a \quad \text{and} \quad y = 7b
\]
where \(a\) and \(b\) are co-prime integers.
### Step 4: Substitute into the product equation
Substituting \(x\) and \(y\) into the product equation gives:
\[
(7a) \times (7b) = 980
\]
This simplifies to:
\[
49ab = 980
\]
Dividing both sides by 49:
\[
ab = \frac{980}{49} = 20
\]
### Step 5: Find pairs of co-prime integers \(a\) and \(b\)
Now we need to find pairs of co-prime integers \(a\) and \(b\) such that their product is 20. The pairs of factors of 20 are:
- (1, 20)
- (2, 10)
- (4, 5)
Among these pairs, we need to identify which ones are co-prime:
- (1, 20) are co-prime.
- (2, 10) are not co-prime.
- (4, 5) are co-prime.
Thus, the valid pairs are (1, 20) and (4, 5).
### Step 6: Calculate the corresponding numbers
Now we can calculate the corresponding numbers for the co-prime pairs:
1. For \(a = 1\) and \(b = 20\):
- \(x = 7 \times 1 = 7\)
- \(y = 7 \times 20 = 140\) (not in the range 20 to 45)
2. For \(a = 4\) and \(b = 5\):
- \(x = 7 \times 4 = 28\)
- \(y = 7 \times 5 = 35\) (both are in the range 20 to 45)
### Step 7: Calculate the sum of the numbers
Now, we can find the sum of the two numbers:
\[
28 + 35 = 63
\]
### Final Answer
The sum of the two numbers is **63**.
