Question 140813
Did you mean that the 'sum of <i>the digits</i> of a two digit number is 10'?  I hope so, otherwise what you said makes no sense at all.


x is the tens digit
y is the ones digit


x + y = 10


The number itself is 10x + y


Subtract 18:  10x + y - 18


If the digits are the same after subtracting 18, then x - 1 = y - 8 => x - y = -7 (if y >=8) or x - 2 = y + (10 - 8) => x - y = 4 (if y < 8)


so, if y >= 8
x - y = -7
x + y = 10

2x = 3 => x not an integer, and y must be less than 8


so
x - y = 4
x + y = 10

2x = 14
x = 7
y = 3

The number is 73.


Check
7 + 3 = 10
73 - 18 = 55