Question

When you reverse the digits of the number 13, the number increases by 18. How many other two-digit numbers increase by 18 when their digits are reversed?

• A. 5
• B. 6
• C. 7
• D. 8
• E. 10

Hint:

The difference between a two-digit number and its reverse is always a multiple of 9. Specifically, \( |(10x+y) - (10y+x)| = 9|x-y| \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A two-digit number can be represented as \( 10x + y \), where \( x \) is the tens digit and \( y \) is the units digit. Reversing the digits gives \( 10y + x \).
Step 2: Key Formula or Approach:
The problem states: \[ (10y + x) - (10x + y) = 18 \] \[ 9y - 9x = 18 \implies y - x = 2 \]
Step 3: Detailed Explanation:
We need to find all pairs \( (x, y) \) such that \( y - x = 2 \), where \( x \in \{1, 2, \dots, 9\} \) and \( y \in \{0, 1, \dots, 9\} \). Possible pairs are:
\( x=1, y=3 \implies 13 \) (Already mentioned in the question)
\( x=2, y=4 \implies 24 \)
\( x=3, y=5 \implies 35 \)
\( x=4, y=6 \implies 46 \)
\( x=5, y=7 \implies 57 \)
\( x=6, y=8 \implies 68 \)
\( x=7, y=9 \implies 79 \)
The question asks for other numbers besides 13. Total pairs found = 7. Other numbers = \( 7 - 1 = 6 \).
Step 4: Final Answer:
There are 6 other such two-digit numbers.