Areteem: Math Challenge ll-A Number Theory

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Math Challenge ll-A Number Theory

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.

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!

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