1. The problem statement, all variables and given/known data (a) Prove that if n is an integer and n2 is a multiple of 3, then n is a multiple of 3. (b) Consider a class of n students. In an exam, the class average is k points. Prove, using contradiction, that at least one student must have received at least k marks in the exam. 2. Relevant equations 3. The attempt at a solution (a) I think I've solved the first one correctly. Here's my attempt. Assume that n is not a multiple of 3. Therefore, n ≠ 3p, where p is an integer. Therefore, n2 ≠ 9p2. Therefore, n2 ≠ 3(3p2). Therefore, n2 is not a multiple of 3. Therefore, by contraposition, if n2 is a multiple of 3, then n is a multiple of 3. (b) The second one I've made a partial attempt as I could not figure how to proceed. Here's my attempt. Assume that class average ≠ k points. Therefore, total sum of marks of all students ≠ nk points. No idea of how to proceed from here onwards. It would be great if you could supply hints on how to carry forward and check my first solution as well.