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Suppose b is any integer. If b mod 12 = 7, what is 5b mod 12? In other words, if division of b by 12 gives a remainder of 7, what is the remainder when 5b is divided by 12? Fill in the blanks to show that the same answer will be obtained no matter what integer is used for b at the start. . Multiply both sides of this equation by 5 and then simplify Because b mod 12 = 7, there is an integer m such that b = 12m + 7 the right-hand side to find values of q and r such that 5b = 12q + r with 0 sr< 12. The result is q = and r = Now 0

Suppose b is any integer. If b mod 12 = 7, what is 5b mod 12? In other words, if division of b by 12 gives a remainder of 7, what is the remainder when 5b is divided by 12? Fill in the blanks to show that the same answer will be obtained no matter what integer is used for b at the start. . Multiply both sides of this equation by 5 and then simplify Because b mod 12 = 7, there is an integer m such that b = 12m + 7 the right-hand side to find values of q and r such that 5b = 12q + r with 0 sr< 12. The result is q = and r = Now 0

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.5: Congruence Of Integers
Problem 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...
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Suppose b is any integer. If b mod 12 = 7, what is 5b mod 12? In other words, if division of b by 12 gives a remainder of 7, what is the remainder when 5b is divided by 12? Fill in the blanks to show that the same answer will be obtained no matter what integer is used for b at the start. Because b mod 12 = 7, there is an integer m such that b = 12m + 7 Multiply both sides of this equation by 5 and then simplify the right-hand side to find values of q and r such that 5b = 12g + r with 0 <r < 12. The result is 9 = and r = Now 0 <r< 12, and q is an integer because products and sums of integers are integers So the uniqueness part of the quotient remainder theorem guarantees that the remainder obtained when 5b is divided by 12 is
Transcribed Image Text:Suppose b is any integer. If b mod 12 = 7, what is 5b mod 12? In other words, if division of b by 12 gives a remainder of 7, what is the remainder when 5b is divided by 12? Fill in the blanks to show that the same answer will be obtained no matter what integer is used for b at the start. Because b mod 12 = 7, there is an integer m such that b = 12m + 7 Multiply both sides of this equation by 5 and then simplify the right-hand side to find values of q and r such that 5b = 12g + r with 0 <r < 12. The result is 9 = and r = Now 0 <r< 12, and q is an integer because products and sums of integers are integers So the uniqueness part of the quotient remainder theorem guarantees that the remainder obtained when 5b is divided by 12 is
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