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Math Challenge ll-A Number Theory
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.
Re: Math Challenge ll-A Number Theory
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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