Question

Exercise 8.25 Suppose a, b, and d are nonzero integers. Suppose dla and d(a2+ 62). 1. Show that dlb2 2. If d is prime, show that dlb. 3. Give an example to show that if d is not prime, then d need not divide b.

Answer 1

Since d/a Therefore a = dm for some integer m. -----(1) and also given d/(a^2 + b^2), hence a^2 + b^2 = dm, for some integer n. or, (dm)^2 + b^2 = dn [by (1)] or d^2 m^2 + b^2 = dn or, b^2 = dn - d^2 m^2 or, b^2 = d(n - dm^2) Since, (n - dm^2) is an integer, it follows that d/d(n - dm^2). Consequently d/b^2 This complete the proof. We have d/b^2, and d is prime. We prove that d/b. If possible let d times b. Then gcd (d, b) = 1, therefore there exist integers x and y so that dx + by = 1 or, b dx + b^2 y = b We have d/b^2 and obviously d/b dx therefore d/(b dx + b^2 y), which implies \r\n d/b, a Contradiction. and therefore we can conclude that, d/b Hence proved Let d = 8, b = 4, then d/b^2 [Because 8/16] but d times b [Because 8 times 4]