To solve this problem, let’s denote the two-digit number as 10a + b, where a is the tens digit and b is the units digit. Based on the information provided, we can establish the following equations:
- The sum of the number and its reverse is 165:
(10a + b) + (10b + a) = 165
This simplifies to:
11a + 11b = 165
Dividing the entire equation by 11 gives:
a + b = 15
- Additionally, we also know that the digits differ by 3:
|a – b| = 3
Now, we have a system of two equations:
- a + b = 15
- |a – b| = 3
From the second equation, we can derive two cases:
Case 1: a – b = 3
In this case, we can express a as:
a = b + 3
Substituting this into the first equation:
(b + 3) + b = 15
Which simplifies to:
2b + 3 = 15
Thus:
2b = 12
b = 6
Substituting back to find a:
a = 6 + 3 = 9
So in this case, we have a = 9 and b = 6, which gives us the number 96.
Case 2: b – a = 3
In this case, we express b as:
b = a + 3
Substituting into the first equation:
a + (a + 3) = 15
This simplifies to:
2a + 3 = 15
So:
2a = 12
a = 6
Then substituting back to find b:
b = 6 + 3 = 9
Thus, we end up with the digits as a = 6 and b = 9, leading to the number 69.
Therefore, the two-digit number that satisfies the conditions of the problem is either 96 or 69. Since both numbers meet the original requirements, you can confirm that:
96 + 69 = 165 and the digits differ by 3.