Question

When the digits of the number 14 are reversed, the number increases by 27. How many other two-digit numbers increase by 27 when their digits are reversed?

• A. 5
• B. 6
• C. 7
• D. 8
• E. 9

Answer 1

The Correct Option is A

To solve the problem of determining how many two-digit numbers increase by 27 when their digits are reversed, let's follow these steps: 

1. Consider a generic two-digit number, represented as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit.
2. When the digits are reversed, the new number becomes \(10y + x\).
3. According to the problem, reversing the digits increases the number by 27. Therefore, we have the equation: \(10y + x = 10x + y + 27\).
4. Simplifying the equation: \(10y + x - 10x - y = 27\)
   \(9y - 9x = 27\)
5. Dividing the entire equation by 9 gives: \(y - x = 3\)
6. This equation means that the units digit \(y\) is 3 more than the tens digit \(x\).
7. To find all possible pairs \((x, y)\) that satisfy this condition, consider:
   • If \(x = 1\), then \(y = 4\) (i.e., 14) - already given in the problem.
   • If \(x = 2\), then \(y = 5\) (i.e., 25).
   • If \(x = 3\), then \(y = 6\) (i.e., 36).
   • If \(x = 4\), then \(y = 7\) (i.e., 47).
   • If \(x = 5\), then \(y = 8\) (i.e., 58).
   • If \(x = 6\), then \(y = 9\) (i.e., 69).
8. Thus, the numbers that increase by 27 when their digits are reversed are: 25, 36, 47, 58, and 69. Excluding 14 from the list since it was initially given for the problem, we are left with these 5 numbers.

Therefore, the correct answer is 5 numbers.