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When you reverse the digits of the numbe...

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

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find how many two-digit numbers increase by 18 when their digits are reversed. We will follow these steps: ### Step 1: Define the two-digit number Let the two-digit number be represented as \( 10x + y \), where \( x \) is the tens digit and \( y \) is the units digit. ### Step 2: Write the reversed number When the digits are reversed, the number becomes \( 10y + x \). ### Step 3: Set up the equation According to the problem, when the digits are reversed, the number increases by 18. Therefore, we can set up the equation: \[ 10y + x = (10x + y) + 18 \] ### Step 4: Simplify the equation Rearranging the equation gives: \[ 10y + x - y - 10x = 18 \] \[ 9y - 9x = 18 \] Dividing the entire equation by 9: \[ y - x = 2 \] ### Step 5: Find possible values for \( x \) and \( y \) From the equation \( y - x = 2 \), we can express \( y \) in terms of \( x \): \[ y = x + 2 \] Since \( x \) and \( y \) are digits, \( x \) can range from 1 to 7 (because \( y \) must be a digit and cannot exceed 9). ### Step 6: List the pairs of \( (x, y) \) Now we can find the pairs of \( (x, y) \): - If \( x = 1 \), then \( y = 3 \) → Number: 13 - If \( x = 2 \), then \( y = 4 \) → Number: 24 - If \( x = 3 \), then \( y = 5 \) → Number: 35 - If \( x = 4 \), then \( y = 6 \) → Number: 46 - If \( x = 5 \), then \( y = 7 \) → Number: 57 - If \( x = 6 \), then \( y = 8 \) → Number: 68 - If \( x = 7 \), then \( y = 9 \) → Number: 79 ### Step 7: Count the valid numbers The valid two-digit numbers that satisfy the condition are: - 13 - 24 - 35 - 46 - 57 - 68 - 79 ### Step 8: Exclude the original number Since the question asks for how many other two-digit numbers increase by 18 when their digits are reversed, we exclude 13 from our count. Thus, the total count of other two-digit numbers is: \[ 6 \] ### Final Answer There are **6 other two-digit numbers** that increase by 18 when their digits are reversed. ---
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