Two units digit of a two-digit number is 3 and seven times the sum of the digits is the number itself. Find the number.

Answer

To solve the problem, we need to find a two-digit number where the units digit is 3, and seven times the sum of the digits equals the number itself. Let's break it down step by step.

### Step 1: Define the digits of the number
Let the tens digit of the two-digit number be represented by \( x \). Since the units digit is given as 3, we can express the two-digit number as:
\[ 10x + 3 \]

**Hint:** Remember that in a two-digit number, the tens place contributes to the value by being multiplied by 10.

### Step 2: Set up the equation based on the problem statement
According to the problem, seven times the sum of the digits equals the number itself. The sum of the digits is:
\[ x + 3 \]
Thus, we can set up the equation:
\[ 7(x + 3) = 10x + 3 \]

**Hint:** Make sure to distribute the 7 correctly across the sum of the digits.

### Step 3: Expand and simplify the equation
Now, let's expand the left side of the equation:
\[ 7x + 21 = 10x + 3 \]

**Hint:** Keep track of the terms on both sides of the equation as you simplify.

### Step 4: Rearrange the equation
Next, we will rearrange the equation to isolate \( x \). We can move \( 7x \) to the right side and 3 to the left side:
\[ 21 - 3 = 10x - 7x \]
This simplifies to:
\[ 18 = 3x \]

**Hint:** When moving terms across the equal sign, remember to change their signs.

### Step 5: Solve for \( x \)
Now, we can solve for \( x \) by dividing both sides by 3:
\[ x = \frac{18}{3} = 6 \]

**Hint:** Division is the inverse operation of multiplication, which helps us isolate \( x \).

### Step 6: Determine the two-digit number
Now that we have \( x = 6 \), we can find the two-digit number:
\[ 10x + 3 = 10(6) + 3 = 60 + 3 = 63 \]

**Hint:** Substitute the value of \( x \) back into the expression for the two-digit number to find the final answer.

### Final Answer
The two-digit number is:
\[ \boxed{63} \]