Three numbers which are coprime to one another are such that the product of the first two is 551 and that of the last two is 1073 . The sum of the three numbers is :

Options

A. 75
B. 81
C. 85
D. 89

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the problem
We have three coprime numbers, let's denote them as \( a \), \( b \), and \( c \). We know that:
- The product of the first two numbers \( a \) and \( b \) is 551.
- The product of the last two numbers \( b \) and \( c \) is 1073.

### Step 2: Set up the equations
From the information given, we can write:
1. \( a \times b = 551 \)
2. \( b \times c = 1073 \)

### Step 3: Find the HCF (which is the common number)
Since \( a \), \( b \), and \( c \) are coprime, the middle number \( b \) is common in both products. Therefore, we need to find the highest common factor (HCF) of 551 and 1073 to determine \( b \).

### Step 4: Factor the numbers
Let's factor both numbers into their prime factors:
- For 551:
- \( 551 = 19 \times 29 \)

- For 1073:
- \( 1073 = 29 \times 37 \)

### Step 5: Identify the common factor
From the prime factorization, we can see that the common factor (HCF) of 551 and 1073 is \( 29 \). Thus, we have:
- \( b = 29 \)

### Step 6: Calculate the other numbers
Now we can find \( a \) and \( c \):
- To find \( a \):
\[ a = \frac{551}{b} = \frac{551}{29} = 19 \]

- To find \( c \):
\[ c = \frac{1073}{b} = \frac{1073}{29} = 37 \]

### Step 7: Calculate the sum of the three numbers
Now that we have all three numbers:
- \( a = 19 \)
- \( b = 29 \)
- \( c = 37 \)

The sum of the three numbers is:
\[ a + b + c = 19 + 29 + 37 = 85 \]

### Final Answer
The sum of the three numbers is **85**.

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