Question 1 (3 points) m n m1 m2 Let r = be a rational number with m E Z, n = N and |m| < n. Let n = P "P2" factorization of n into (positive) primes, where the P1, P2, Pr are distinct and M1, M2, ..., Mr > 0 (if n = 1, this is the "empty product", but in this case m = 0). Show that there are integers ij such that mr Pr be the and |aj| < Pi r mi r = = 11 12 + P1 1m1 21 armr + + + Mr m1 2 P1 P2 Pr i=1 j=1 (Hint: Let P, Q be coprime natural numbers, and let m be an integer such that |m| < PQ. Show that there are integers a, with |a| < Q and || < P such that m = m1 n P + BQ. = conclude that r = + Now do a suitable induction and treat by If P = p and Q P' repeated division with remainder.) Q P P