Answer: The sum is 127
Step-by-step explanation:
A 2-digit number N = ab can be written as (where a and b are single-digit numbers)
a*10 + b.
Now, we want that:
(a + 2)*(b + 2) = a*10 + b.
So we must find all the solutions to that equation such that a can not be zero (if a = 0, then the number is not a 2-digit number)
We have:
(a + 2)*(b + 2) = a*b + 2*a + 2*b + 4 = a*10 + b
a*b + 2*b - b + 4 = a*10 - a*2
a*b + 4 + b = a*8
a*b + 4 + b - a*8 = 0.
Now we can give one of the variables different values, and see if the equation has solutions:
>a = 1:
1*b + 4 + b - 8 = 0
2*b - 4 = 0
b = 4/2 = 2
Then the number 12 has the property.
> if a = 2:
2*b + 4 + b -16 = 0
3b -12 = 0
b = 12/3 = 4
The number 24 has the property.
>a = 3 is already known, here the solution is 35.
>a = 4.
4*b + 4 + b - 8*4 = 0
5*b + 4 - 32 = 0
5*b = 28
b = 28/5
this is not an integer, so here we do not have a solution.
>if a = 5.
5*b + 4 + b - 8*5 = 0
6b + 4 - 40 = 0
6b - 36 = 0
b = 36/6 = 6
So the number 56 also has the property.
>if a = 6
6*b + 4 + b - 8*6 = 0
7b + 4 - 48 = 0
7b - 44 = 0
b = 44/7 this is not an integer, so here we do not have any solution.
>if a = 7
7*b + 4 + b -8*7 = 0
8b -52 = 0
b = 52/8 = 6.5 this is not an integer, so we here do not have a solution.
>if a = 8
8*b + 4 + b -8*8 = 0
9*b + 4 - 64 = 0
9*b = 60
b = 60/9 this is not an integer, so we here do not have any solution:
>if a = 9
9*b + 4 + b - 8*9 = 0
10b + 4 - 72 = 0
10b -68 = 0
b = 68/10 again, this is not an integer.
So the numbers with the property are:
12, 24, 35 and 56
And the sum is:
12 + 24 + 35 + 56 = 127
