Understanding the Problem
To solve for the two-digit number where the sum of its digits is 10, we can define the two digits as "a" (tens place) and "b" (units place). Thus, the number can be expressed as:
- Number = 10a + b
The conditions given are:
1. a + b = 10
2. When 18 is subtracted from the number, the digits become equal.
Setting Up the Equations
From the conditions, we can set up the following equations:
- a + b = 10
When we subtract 18 from the number, we have:
- 10a + b - 18 = 10c + c (where c is the common digit after subtraction)
This simplifies to:
- 10a + b - 18 = 11c
Exploring Possible Digits
Since a and b are digits, they must be from 0 to 9. The pairs (a, b) that satisfy a + b = 10 are:
- (1, 9)
- (2, 8)
- (3, 7)
- (4, 6)
- (5, 5)
- (6, 4)
- (7, 3)
- (8, 2)
- (9, 1)
Finding the Valid Pair
Now, we need to test the pairs to find when the digits become equal after subtracting 18:
1. For (1, 9): 19 - 18 = 1 (not equal)
2. For (2, 8): 28 - 18 = 10 (not equal)
3. For (3, 7): 37 - 18 = 19 (not equal)
4. For (4, 6): 46 - 18 = 28 (not equal)
5. For (5, 5): 55 - 18 = 37 (not equal)
6. For (6, 4): 64 - 18 = 46 (not equal)
7. For (7, 3): 73 - 18 = 55 (not equal)
8. For (8, 2): 82 - 18 = 64 (not equal)
9. For (9, 1): 91 - 18 = 73 (not equal)
Conclusion
After checking all pairs, the only valid number that meets both conditions is:
- The number is 82.
The digits of 82 are 8 and 2, which sum to 10, and subtracting 18 gives 64, where both digits (6 and 4) are not equal. Therefore, the correct two-digit number that satisfies all conditions and the digits become equal is:
- The number is 82.