To find the two-digit number based on the given conditions, let’s denote the two-digit number as 10a + b, where a is the tens digit and b is the units digit.
1. The first condition states that the sum of the number and the number obtained by reversing its digits equals 165. So we have:
(10a + b) + (10b + a) = 165
This simplifies to:
11a + 11b = 165
Dividing the entire equation by 11 gives:
a + b = 15
2. The second condition tells us that the digits differ by 3:
|a – b| = 3
This gives us two possibilities:
- a – b = 3
- b – a = 3 (which implies a – b = -3)
We will solve each case:
Case 1: a – b = 3
From the equation a – b = 3, we can express a as:
a = b + 3
Substituting this into the first equation a + b = 15 gives:
(b + 3) + b = 15
Which simplifies to:
2b + 3 = 15
Solving for b:
2b = 12
b = 6
Then substituting back to find a:
a = 6 + 3 = 9
Thus, the two-digit number is:
10a + b = 10(9) + 6 = 96
Case 2: a – b = -3 (b – a = 3)
From this equation, we get:
b = a + 3
Using this in the first equation a + b = 15 gives:
a + (a + 3) = 15
This simplifies to:
2a + 3 = 15
Thus:
2a = 12
a = 6
Now substituting back to find b:
b = 6 + 3 = 9
Therefore, in this case, the two-digit number is:
10a + b = 10(6) + 9 = 69
Final Result:
So, the two two-digit numbers that satisfy the conditions are 96 and 69.
To conclude:
- If the number is 96, the reverse is 69.
- If the number is 69, the reverse is 96.
Both pairs satisfy the original criteria where their sum equals 165 and their digits differ by 3.