Prove the Chinese Remainder Theorem: Let
Furthermore, any two solutions
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Chapter 2 Solutions
Elements Of Modern Algebra
- 31. Prove statement of Theorem : for all integers and .arrow_forwardLet and be positive integers. If and is the least common multiple of and , prove that . Note that it follows that the least common multiple of two positive relatively prime integers is their product.arrow_forwardLabel each of the following statement as either true or false. Let and be integers with. Then if and only if the remainder in the Division Algorithm, when is divided by, isarrow_forward
- Prove that the statements in Exercises 116 are true for every positive integer n. a+ar+ar2++arn1=a1rn1rifr1arrow_forwardProve that if and are real numbers such that , then there exist a rational number such that . (Hint: Use Exercise 25 to obtain such that . Then choose to be the least integer such that . With these choices of and , show that and then that .) If and are positive real numbers, prove that there exist a positive integer such that . This property is called Archimedean Property of the real numbers. (Hint: If for all , then is an upper bound for the set . Use the completeness property of to arrive at a contradiction.)arrow_forward25. Prove that if and are integers and, then either or. (Hint: If, then either or, and similarly for. Consider for the various causes.)arrow_forward
- Find the smallest integer in the given set. { and for some in } { and for some in }arrow_forwardAssume the statement from Exercise 30 in section 2.1 that for all and in . Use this assumption and mathematical induction to prove that for all positive integers and arbitrary integers .arrow_forward23. Find all distinct principal ideals of for the given value of . a. b. c. d. e. f.arrow_forward
- 30. Prove statement of Theorem : for all integers .arrow_forwardLet x and y be integers, and let m and n be positive integers. Use mathematical induction to prove the statements in Exercises 1823. ( The definitions of xn and nx are given before Theorem 2.5 in Section 2.1 ) (m+n)x=mx+nxarrow_forwardLet and be integers, and let and be positive integers. Use mathematical induction to prove the statements in Exercises. The definitions of and are given before Theorem in Sectionarrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage