Question

A two digit number is such that the unit digit mu tiplied by 3 is equal to three more than the sum of the digits. When the digits are reversed the resulting number is 18 less than the original number. Find the units digit of the original number.

• A. 8
• B. 5
• C. 4
• D. 2
• E. 1

Answer 1

The Correct Option is B

Step 1: Understand the problem.
We are given a two-digit number where the unit digit multiplied by 3 is equal to three more than the sum of the digits. Additionally, when the digits are reversed, the resulting number is 18 less than the original number.

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 based on the conditions.
1. The unit digit multiplied by 3 is equal to three more than the sum of the digits:
\( 3b = a + b + 3 \)
Simplifying the equation:
\( 3b - b = a + 3 \)
\( 2b = a + 3 \)
\( a = 2b - 3 \) (Equation 1)

2. When the digits are reversed, the resulting number is 18 less than the original number:
\( 10b + a = 10a + b - 18 \)
Simplifying the equation:
\( 10b - b = 10a - a - 18 \)
\( 9b = 9a - 18 \)
\( b = a - 2 \) (Equation 2)

Step 3: Solve the system of equations.
Substitute \( a = 2b - 3 \) from Equation 1 into Equation 2:
\( b = (2b - 3) - 2 \)
Simplifying:
\( b = 2b - 5 \)
\( b - 2b = -5 \)
\( -b = -5 \)
\( b = 5 \)

Step 4: Conclusion.
The units digit of the original number is 5.

Final Answer:
The correct option is (B): 5.