Q1. Two statements are given below. For each, an erroneous proof is provided. There is one main error in each proof. State what this error is and explain why it is an error. (Note that both of these statements are false, but this is irrelevant to the question and your answer.) (a) Statement: For all r E Z, if 2r + 3 = 11 (mod 4), then 4 | r. Proof: Let r E Z. Assume that 2r + 3 = 11 (mod 4). Therefore, using Congruence Add and Multiply, we obtain 2r + 3 =11 (mod 4) 2r = 8 (mod 4) r=4 (mod 4) 1=0 (mod 4) Therefore, by the definition of congruence, we have that 4 | (x - 0), which gives 4 | r, as desired. O (b) Statement: For all integers a and y, if (r, y) is an integer solution to -3r + 7y = 21, then a + y = 3 (mod 4). Proof: By inspection, we can see that r = 0 and y = 3 is a solution to -3x + 7y = 21. By Linear Diophantine Equation Theorem Part 2 (LDET 2), the general solution is {(x, y) : x = 7n,y = 3 - 3n, ne Z). Therefore, if (, y) is an integer solution to -3r+7y = 21, then cty = 7n+3-3n = 3+4n for some n E Z. Therefore, r + y = 3 (mod 4). 0