In the solution, it states that 59 and 45 have a difference of 7. This is clearly wrong. It should have stated "For choice ii.), 59 and 45 have a difference of 14, which is a multiple of 7, so this is the correct answer."

@Bella Both are actually correct. I believe Professor Loh first found 27 (mod 4) to be 3, and then doubled this to get 6. Professor Loh is emphasizing the simplification step, which makes working with larger numbers easier. And if we simplify this further, 6 is 2 (mod 4). So your calculator is not wrong; 27 + 27 is 2 (mod 4). But noting Professor Loh's strategy will help you solve these types of problems more efficiently 🙂