By Jean A. Dieudonne, J.B. Carrell

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**Example text**

17. If pis prime and (a, 18. b) =p, then (a2, il) =? Prove or disprove each of the following statements: (a) If pis prime andpI (a2 + b2) andpI (c2 + Jl-), then p I (a2 - c2). (b) If pis prime andpI(a2 + b 2) andpI (c2 + t:P), then p I (a2 + c2). (c) If pis prime andpI aandpI (a2 + il), thenpI b. f B. 19. ' P'i [If, where p1, Pz, ... , Pk are distinct positive primes and each r1, s1 2: 0. Prove that aI b if and only if r1 s s1 for every i. · • · · · · p',; and b fl'i'P'i� 20. If a p�'P'iP? ] 21.

23. : 1. * 24. Let a, b, c Z. Prove that the equation E ax +by == c has integer solutions if (a, b) I c. and only if 25. (a) If a, b, u, v E Z are such that (b) Show by example that if au +bv = 1, prove that (a, b) = 1. au+ bv = d > 1, then (a, b) may not bed. 26. If a I c and b I c and (a, b) = d, prove that ab Ied. 27. If c lab and (c, a)= d, prove that cldb. 28. Prove that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. ] 29. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9.

Hence m must have positive divi sors other than 1 or m, say m = ab with 1< a< m and 1< b< m. Since both a and bare less than m (the smallest element of S), neither a nor bis in S. By the definition of S, both a and bare the product of primes , say a= P1P2 · • ·p , and with r;;::; 1, s;;::; 1, and each p1, (/jprime. Therefore is a product of primes, so that m it S. We have reached a contradiction: m E S and m it S. Therefore, S must be empty. com). 10, which is proved on page 21, to do the factorization.