Answer :
To solve this problem, we need to determine the three-digit number based on the given conditions. Let's denote the number as [tex]\( \overline{abc} \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the digits of the number.
### Step-by-Step Solution
1. Sum of the digits:
The sum of the digits [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] is given as 8.
[tex]\[ a + b + c = 8 \][/tex]
2. Middle digit relation:
The middle digit [tex]\( b \)[/tex] is three times the sum of the other two digits [tex]\( a \)[/tex] and [tex]\( c \)[/tex].
[tex]\[ b = 3(a + c) \][/tex]
3. Equality of the number and its reverse:
The number is the same as the number obtained by reversing the order of the digits. Hence, [tex]\( \overline{abc} = \overline{cba} \)[/tex].
This means:
[tex]\[ 100a + 10b + c = 100c + 10b + a \][/tex]
Simplifying this equation:
[tex]\[ 99a = 99c \implies a = c \][/tex]
4. Substitute [tex]\( a = c \)[/tex] into the previous equations:
Since [tex]\( a = c \)[/tex], we can use this relationship in our original equations.
[tex]\[ a + b + c = 8 \implies a + b + a = 8 \implies 2a + b = 8 \][/tex]
[tex]\[ b = 3(a + c) \implies b = 3(2a) \implies b = 6a \][/tex]
5. Substitute [tex]\( b = 6a \)[/tex] into [tex]\( 2a + b = 8 \)[/tex]:
[tex]\[ 2a + 6a = 8 \implies 8a = 8 \implies a = 1 \][/tex]
6. Determine values of [tex]\( b \)[/tex] and [tex]\( c \)[/tex]:
Since [tex]\( a = c \)[/tex], we have [tex]\( a = 1 \)[/tex] and [tex]\( c = 1 \)[/tex].
[tex]\[ b = 6a \implies b = 6 \times 1 = 6 \][/tex]
So the digits of the number are [tex]\( a = 1 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = 1 \)[/tex]. Therefore, the three-digit number is 161.
### Conclusion
The three-digit number that satisfies all the given conditions is:
[tex]\[ \boxed{161} \][/tex]
### Step-by-Step Solution
1. Sum of the digits:
The sum of the digits [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] is given as 8.
[tex]\[ a + b + c = 8 \][/tex]
2. Middle digit relation:
The middle digit [tex]\( b \)[/tex] is three times the sum of the other two digits [tex]\( a \)[/tex] and [tex]\( c \)[/tex].
[tex]\[ b = 3(a + c) \][/tex]
3. Equality of the number and its reverse:
The number is the same as the number obtained by reversing the order of the digits. Hence, [tex]\( \overline{abc} = \overline{cba} \)[/tex].
This means:
[tex]\[ 100a + 10b + c = 100c + 10b + a \][/tex]
Simplifying this equation:
[tex]\[ 99a = 99c \implies a = c \][/tex]
4. Substitute [tex]\( a = c \)[/tex] into the previous equations:
Since [tex]\( a = c \)[/tex], we can use this relationship in our original equations.
[tex]\[ a + b + c = 8 \implies a + b + a = 8 \implies 2a + b = 8 \][/tex]
[tex]\[ b = 3(a + c) \implies b = 3(2a) \implies b = 6a \][/tex]
5. Substitute [tex]\( b = 6a \)[/tex] into [tex]\( 2a + b = 8 \)[/tex]:
[tex]\[ 2a + 6a = 8 \implies 8a = 8 \implies a = 1 \][/tex]
6. Determine values of [tex]\( b \)[/tex] and [tex]\( c \)[/tex]:
Since [tex]\( a = c \)[/tex], we have [tex]\( a = 1 \)[/tex] and [tex]\( c = 1 \)[/tex].
[tex]\[ b = 6a \implies b = 6 \times 1 = 6 \][/tex]
So the digits of the number are [tex]\( a = 1 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = 1 \)[/tex]. Therefore, the three-digit number is 161.
### Conclusion
The three-digit number that satisfies all the given conditions is:
[tex]\[ \boxed{161} \][/tex]
