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Grace remembers that if they wanted to buy some number of $91 items, spending 77 times the amount of money Owed would return $14 in change. That is, 77x 14 (mod 91). Leah remembers that if they wanted to buy some number of $28 items, spending 12 times the amount of money wed would return $8 in change. That is, 12x 8 (mod 28). a) Find a single value of a that solves both congruences. b) Grace says, "I found a value of that solves both congruences, but it took a stupendously long time using trial ar error." Leah gloats, "I was able to find a value of a without using any trial and error!" Briefly describe how Leah might have found their answer to part (a)

Grace remembers that if they wanted to buy some number of $91 items, spending 77 times the amount of money Owed would return $14 in change. That is, 77x 14 (mod 91). Leah remembers that if they wanted to buy some number of $28 items, spending 12 times the amount of money wed would return $8 in change. That is, 12x 8 (mod 28). a) Find a single value of a that solves both congruences. b) Grace says, "I found a value of that solves both congruences, but it took a stupendously long time using trial ar error." Leah gloats, "I was able to find a value of a without using any trial and error!" Briefly describe how Leah might have found their answer to part (a)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section: Chapter Questions
Problem 29RE
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(c) Suppose x and y are both solutions to the pair of congruences. What is the smallest possible positive value of x - y? (d) Briefly explain your answer to part (c).
Transcribed Image Text:(c) Suppose x and y are both solutions to the pair of congruences. What is the smallest possible positive value of x - y? (d) Briefly explain your answer to part (c).
Grace owes Leah some money, but neither of them can remember how much! Let $x be the amount of money owed, where x is some positive integer. Grace remembers that if they wanted to buy some number of $91 items, spending 77 times the amount of money owed would return $14 in change. That is, 77x 14 (mod 91). Leah remembers that if they wanted to buy some number of $28 items, spending 12 times the amount of money owed would return $8 in change. That is, 12x = 8 (mod 28). (a) Find a single value of that solves both congruences. (b) Grace says, "I found a value of a that solves both congruences, but it took a stupendously long time using trial and error." Leah gloats, "I was able to find a value of x without using any trial and error!" Briefly describe how Leah might have found their answer to part (a).
Transcribed Image Text:Grace owes Leah some money, but neither of them can remember how much! Let $x be the amount of money owed, where x is some positive integer. Grace remembers that if they wanted to buy some number of $91 items, spending 77 times the amount of money owed would return $14 in change. That is, 77x 14 (mod 91). Leah remembers that if they wanted to buy some number of $28 items, spending 12 times the amount of money owed would return $8 in change. That is, 12x = 8 (mod 28). (a) Find a single value of that solves both congruences. (b) Grace says, "I found a value of a that solves both congruences, but it took a stupendously long time using trial and error." Leah gloats, "I was able to find a value of x without using any trial and error!" Briefly describe how Leah might have found their answer to part (a).
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