MATH220 December 2011
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q3 • Q4 • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q8 •
Prove that if p is a prime number and p > 4, then
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!
Note: this hint was giving on the exam itself.
- Think about the possible remainders when p is divided by 6.
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By saying that we look at primes larger than 4, what we really mean is we look at all the primes except 2 and 3 (which makes sense since 22 = 4 and 32 = 9 which is 3 mod 6; so these two don't work anyway).
Now any prime p other than 2 and 3 will then clearly NOT be a multiple of two and NOT be a multiple of 3. This means that
since integers that are 0, 2 or 4 mod 6 are all even and since an integer that is 3 mod 6 is a multiple of 3.
we can conclude that if p is a prime larger than 4, then its square is always 1 mod 6.
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