Explanation Given information: | a | = r s , ( r , s ) = 1 Formula used: | n a | = | a | ( | a | , n ) Calculation: Let ( r , s ) = 1 imply that r x + s y = 1 for some integer x and y . In this linear combination r x + s y = 1 , ( r , y ) = 1 and ( s , x ) = 1 . Now, a = 1 ⋅ a a = ( r x + s y ) a a = r x a + s y a a = s y a + r x a a = b 1 + b 2 where b 1 = s y a , b 2 = r x a | b 1 | = | s y a | = | a | ( | a | , s y

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Chapter 4.7, Problem 21E

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To prove: If r and s are relatively prime, then a can be written in the form a=b1+b2, where b1 has order r and b2 has order s.

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Chapter 4.7, Problem 20E

Chapter 4.7, Problem 22E

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To prove: If r and s are relatively prime, then a can be written in the form a = b 1 + b 2 , where b 1 has order r and b 2 has order s .

Solution Summary: The author explains how a can be written in the form of (r,s)=1 if r and s are relatively prime.

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To prove: If r and s are relatively prime, then a can be written in the form a=b1+b2, where b1 has order r and b2 has order s.

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Chapter 4 Solutions