Understanding the Problem
We need to find a two-digit number where:
- The product of its digits equals 18.
- When the number is subtracted from the number obtained by reversing its digits, the result is equal to the original number.
Representing the Number
Let the two-digit number be represented as "10a + b", where:
- a = the digit in the tens place
- b = the digit in the units place
Setting Up the Equations
1. We know the product of the digits:
- a * b = 18
2. When we reverse the digits, the new number is "10b + a". According to the problem, when we subtract the original number from the reversed number, we have:
- (10b + a) - (10a + b) = 10b + a - 10a - b
- This simplifies to: 9b - 9a = 9(b - a) = (10a + b)
Finding Possible Values
- From the product equation a * b = 18, we list the pairs:
- (2, 9), (3, 6), (6, 3), (9, 2)
- Next, we check if any of these pairs satisfy the subtraction condition.
Checking the Pairs
1. For (2, 9):
- Number = 29; Reversed = 92
- 92 - 29 = 63 (not valid)
2. For (3, 6):
- Number = 36; Reversed = 63
- 63 - 36 = 27 (not valid)
3. For (6, 3):
- Number = 63; Reversed = 36
- 36 - 63 = -27 (not valid)
4. For (9, 2):
- Number = 92; Reversed = 29
- 29 - 92 = -63 (not valid)
Conclusion
After checking all pairs, the only valid number fitting both conditions is:
The number is 63.
This solution meets the requirement of having a product of digits equal to 18 and satisfies the condition of digit reversal.