This problem explores an attack on RSA with small private key d. Preliminaries. Let r be a positive rational number and write r a/b with a, b e N. Let 90,91,...9m N be the sequence of quotients produced by applying the Euclidean Algorithm to the numerator and denominator of r: a b To Tm-3 = 9m-15m-2 +Tm-1, Tm-2 = 9mm-1 + rm, Recall the familiar sequences T- < = gob + ro, = qiro + r, = 92r1 + 12, A-2 = 0, A-1 = 1, A B-2 = 1, B-1 = 0, B = B 2B The quotients A/B (0 i m) are called the convergents of r because they oscillate around and converge toward r as i increases, with Am = a, Bm = b, and hence Am/Bm = r. In fact, the following theorem (which you may use without proof) asserts that any rational number sufficiently close to r must occur as one of the convergents: 0