This is the technique I prefer. It's sometimes called "guess and check" but I don't like this term. It's finding an intelligent approximation to the answer and then modifying the approximation if necessary.

Half of 123 is 62.5 so if one of the numbers were 61 and the other were 62 the sum would be 123 but the difference would only be 1.

The difference needs to be 45, that is 44 more than 1 so increase one of 61 and 62 by 22 and decrease the other by 22. The sum is still be 123 and the difference might be 45. Try it. Is the difference 45? If not how would you modify what you did to get a difference of 45?