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By Charles L. Dodgson

ISBN-10: 141817985X

ISBN-13: 9781418179854

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Consequently, our result is established in all cases. 2 1. Prove that if a and b are integers, with b > 0, then there exist unique integers q and r satisfying a= qb + r, where 2b ::S r < 3b. 2. Show that any integer of the form 6k + 5 is also of the form 3 j + 2, but not conversely. 3. Use the Division Algorithm to establish the following: (a) The square of any integer is either of the form 3k or 3k + 1. (b) The cube of any integer has one of the forms: 9k, 9k + 1, or 9k + 8. (c) The fourth power of any integer is either of the form 5k or 5k + 1.

8. Prove the following: (a) The sum of the squares of two odd integers cannot be a perfect square. (b) The product of four consecutive integers is lless than a perfect square. 9. Establish that the difference of two consecutive cubes is never divisible by 2. 10. Foranonzerointegera, showthatgcd(a, 0) =I a l,gcd(a, a)= Ia l,andgcd(a, 1) = 1. 11. If a and b are integers, not both of which are zero, verify that gcd(a, b)= gcd(-a, b)= gcd(a, -b)= gcd(-a, -b) 12. Prove that, for a positive integer n and any integer a, gcd(a, a+ n) divides n; hence, gcd(a, a+ 1) = 1.

D) The only prime p for which 3p + 1 is a perfect square is p = 5. (e) The only prime of the form n 2 - 4 is 5. 4. If p :::: 5 is a prime number, show that p 2 + 2 is composite. ] 5. (a) Given that p is a prime and p I an, prove that pn I an. (b) If gcd(a, b) = p, a prime, what are the possible values of gcd(a 2 , b 2 ), gcd(a 2 , b) and gcd(a 3 , b 2 )? 6. Establish each of the following statements: (a) Every integer of the form n 4 + 4, with n > 1, is composite. ] (b) If n > 4 is composite, then n divides (n - 1)!.

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An elementary treatise on determinants by Charles L. Dodgson

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