Three numbers which are co-prime to one an- other are such that the product of the first two is 475 and that of the last two numbers is 1075. Find the sum of the three numbers.

Options

A. 75
B. 78
C. 87
D. 89

Answer

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

### Step 1: Define the Variables
Let the three co-prime numbers be \( A \), \( B \), and \( C \).

### Step 2: Set Up the Equations
From the problem, we know:
- \( A \times B = 475 \) (Equation 1)
- \( B \times C = 1075 \) (Equation 2)

### Step 3: Prime Factorization
We will perform the prime factorization of both 475 and 1075.

**Prime Factorization of 475:**
- Divide by 5: \( 475 \div 5 = 95 \)
- Divide by 5 again: \( 95 \div 5 = 19 \)
- So, \( 475 = 5^2 \times 19 \).

**Prime Factorization of 1075:**
- Divide by 5: \( 1075 \div 5 = 215 \)
- Divide by 5 again: \( 215 \div 5 = 43 \)
- So, \( 1075 = 5^2 \times 43 \).

### Step 4: Identify Common Factors
From the factorization, we see that:
- From Equation 1: \( A \times B = 5^2 \times 19 \)
- From Equation 2: \( B \times C = 5^2 \times 43 \)

Here, \( B \) is common in both equations.

### Step 5: Express \( A \) and \( C \) in terms of \( B \)
Since \( B \) is common, we can express \( A \) and \( C \) as:
- From Equation 1: \( A = \frac{475}{B} = \frac{5^2 \times 19}{B} \)
- From Equation 2: \( C = \frac{1075}{B} = \frac{5^2 \times 43}{B} \)

### Step 6: Determine the Value of \( B \)
Since \( A \), \( B \), and \( C \) are co-prime, \( B \) must be a factor that allows \( A \) and \( C \) to remain co-prime. The only suitable value for \( B \) that satisfies both equations and keeps \( A \) and \( C \) co-prime is \( B = 25 \).

### Step 7: Calculate \( A \) and \( C \)
Now we can find \( A \) and \( C \):
- \( A = \frac{475}{25} = 19 \)
- \( C = \frac{1075}{25} = 43 \)

### Step 8: Find the Sum of the Three Numbers
Now we can find the sum:
\[ A + B + C = 19 + 25 + 43 \]

Calculating the sum:
\[ 19 + 25 = 44 \]
\[ 44 + 43 = 87 \]

Thus, the sum of the three numbers is \( 87 \).

### Final Answer
The sum of the three numbers is \( 87 \).

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