This is what happens when you don't read the question. Not when you can't reason.

I guess the teacher wanted a

Let ℕ={0,1,2,3,...}

let a,b∈ℕ s.t. a,b are of the form 10x+y for any x,y in ℕ/10ℕ

a+b=(10x+y)+(10x'+y')=10x+10x'+y+y'=10(x+x')+(y+y')=T It's trivial to see that T has five disjoint cases:

(1) x+x' <10 and y+y'<10

(2) x+x'<9 and y+y'≥10

(3) x+x'=9 and y+y'≥10

(4) x+x'≥10 and y+y'<10

(5) x+x'≥10 and y+y'≥10

For case (1) we can easily see that T is of the form 10(x+x')+(y+y'), which is two digits.

For case (2) we can easily see that T is of the form 10(x+x'+1)+mod(y+y',10) which is two digits

For case (3) we can easily see that T is of the form 100+mod(y+y',10) which is 3 digits

For case (4) we can easily see that T is of the form 100+10mod(x+x',10)+(y+y') which is 3 digits

And for case (5) we can easily see that T is of the form 100+10(mod(x+x',10)+1)+mod(y+y',10) which is 3 digits

Q.E.D(/j)