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Math Challenge ll-A Number Theory

 
 
LiangNeo的头像
Math Challenge ll-A Number Theory
LiangNeo - 2019年06月15日 Saturday 20:49
 

These problems are in Lecture 5. I need the solution for problems 5.21 a  and 5.21 b  because the solution doesn't make sense.

For part a, the solution says "We see that the set of distinct values of n · 5 is {0, 1, 2, 3, 4, 5, 6, 7}, whereas the set of distinct values of n · 8 is {0, 2, 4, 6}. Note that gcd(5,8)=1, whereas gcd(6,8)=2 > 1." This solution doesn't make sense to the question.

For part b, the solution says "Each number has its own inverse."   This solution is irrelevant to the question.

 
ProfessorAreteem的头像
Re: Math Challenge ll-A Number Theory
ProfessorAreteem - 2019年06月17日 Monday 19:40
 
There seemed to be a problem uploading the solution to those questions, as they are missing the first half. Here are the full solutions:


  • For 21(a): We have that $n \cdot 5 = 0, 5, 2, 7, 4, 1, 6, 3 \pmod{8}$, and $n \cdot 6 = 0, 6, 4, 2, 0, 6, 4, 2 \pmod{8}$. We see that the set of distinct values of $n \cdot 5$ is $\{0,1,2,3,4,5,6,7\}$, whereas the set of distinct  values of $n \cdot 6$ is $\{0, 2, 4, 6\}$. Note that $\gcd(5,8) = 1$, whereas $\gcd(6,8) = 2 >1$. 
  • For 21(b): The numbers that are relatively prime to $8$ are $1,3,5,7$. We see $1^2 \equiv 3^2 \equiv 5^2 \equiv 7^2 \equiv 1 \pmod{8}$. Each number is their own inverse.

Solutions on the class page have also been updated. Thanks for letting us know!