Science:Math Exam Resources/Courses/MATH312/December 2005/Question 03

MATH312 December 2005

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Question 03
Multiple Choice The number of primitive roots of 98 is (a) 0 (b) 12 (c) 34 (d) 42 (e) None of the above.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The number of primitive roots modulo n is equal to $\displaystyle \phi (n)$ provided that $n\in \{2,4,p^{k},2p^{k}\}$ for some odd prime p.

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The answer is d = 42 We use the multiplicativity of the phi function to see that $\displaystyle \phi (98)=\phi (2\cdot 49)=\phi (2\cdot 7^{2})=\phi (2)\phi (7^{2})=7(7-1)=42$ As the phi function computes the number of primitive roots modulo n, we are done.

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