if the sum of the digit of a two digit number is 9 . While their product is 14. what is the number.

Options

A. 36
B. 63
C. 54
D. 72

Answer

To solve the problem step by step, we need to find a two-digit number where the sum of its digits is 9 and the product of its digits is 14.

### Step 1: Define the digits
Let the two-digit number be represented as \(10a + b\), where \(a\) is the tens digit and \(b\) is the units digit.

### Step 2: Set up the equations
From the problem, we have two conditions:
1. The sum of the digits:
\[ a + b = 9 \]
2. The product of the digits:
\[ a \times b = 14 \]

### Step 3: Solve the equations
We can express \(b\) in terms of \(a\) using the first equation:
\[ b = 9 - a \]

Now, substitute \(b\) into the second equation:
\[ a \times (9 - a) = 14 \]

### Step 4: Expand and rearrange the equation
Expanding the equation gives:
\[ 9a - a^2 = 14 \]
Rearranging it leads to:
\[ a^2 - 9a + 14 = 0 \]

### Step 5: Factor the quadratic equation
Next, we need to factor the quadratic equation:
\[ (a - 7)(a - 2) = 0 \]

### Step 6: Solve for \(a\)
Setting each factor to zero gives us:
1. \(a - 7 = 0 \Rightarrow a = 7\)
2. \(a - 2 = 0 \Rightarrow a = 2\)

### Step 7: Find corresponding \(b\) values
Now we can find the corresponding \(b\) values for each \(a\):
- If \(a = 7\):
\[ b = 9 - 7 = 2 \]
- If \(a = 2\):
\[ b = 9 - 2 = 7 \]

### Step 8: Form the two-digit numbers
The two-digit numbers formed are:
1. From \(a = 7\) and \(b = 2\): \(72\)
2. From \(a = 2\) and \(b = 7\): \(27\)

### Step 9: Check the conditions
Now we check both numbers against the conditions:
- For \(72\):
- Sum: \(7 + 2 = 9\)
- Product: \(7 \times 2 = 14\)
- For \(27\):
- Sum: \(2 + 7 = 9\)
- Product: \(2 \times 7 = 14\)

Both numbers satisfy the conditions, but since the question asks for a two-digit number, we can conclude that the valid answers are \(72\) and \(27\).

### Final Answer
The two-digit numbers that satisfy the conditions are \(72\) and \(27\).