Question 2. 2.1: Most got this wrong. Many gave an example where the proof fails, but did not identify the flaw in the "inductive proof". 2.2: Most students could not finish the inductive step, some mixed up variable names 'n' and 'k' in the inductive step. Question 3. 3.1: most got this right, solved it numerically. Some got the algorithm for GCD right, but didn't know what number was the GCD. 3.2: most either got it entirely or did not write anything relevant. Question 4. Most students did well. Question 5. 5.1: Few students got the second part: (n+m)/2 lies between n and m and therefore is composite. Most got the first part. 5.2: Didn't know how to check only for prime divisors <= sqrt(number). Didn't show result of division. You were asked to show all work. Question 6. 6.1: Some students used properties of congruences (not allowed by the question), and some tried to work backwards from the conclusion to be proved, without clarifying what they were doing. 6.2: Didn't write what arithmetic rules you were using. Didn't simplify a number n to the smallest one congruent to it modulo 19, and therefore had to do long computations.