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# Need help with a mod problem watch

1. Show that 1/1 + 1/2 + 1/3 + 1/4 +1/5 +1/6 ... 1/p-1 = 1 + 2 + 3 + 4 + 5 + 6 + ... + p-1 (modp)

When i asked my friend he said "it's obvious" because each non zero integer (modp) is equivalent to the reciprocal of one and only one non-zero integer (modp), but I don't get why that is the case.
2. (Original post by CancerousProblem)
Show that 1/1 + 1/2 + 1/3 + 1/4 +1/5 +1/6 ... 1/p-1 = 1 + 2 + 3 + 4 + 5 + 6 + ... + p-1 (modp)

When i asked my friend he said "it's obvious" because each non zero integer (modp) is equivalent to the reciprocal of one and only one non-zero integer (modp), but I don't get why that is the case.
Pick a prime e.g. 7 then find mod ) what can you see about the inverses? Can you apply this more generally?

Or pick a non-prime e.g. 6 what is ? Why does this happen?
3. (Original post by tombayes)
Pick a prime e.g. 7 then find mod ) what can you see about the inverses? Can you apply this more generally?

Or pick a non-prime e.g. 6 what is ? Why does this happen?
I can see the inverses are a rearrangement of 1, 2 ,3 4... (p-1) but I can't see any reason this should be the case for any prime p.

Also I have no idea what you mean by 3^-1 mod6, is that even valid? That makes no sense...
4. thanks for helping, but I still don't get it
5. (Original post by CancerousProblem)
Also I have no idea what you mean by 3^-1 mod6, is that even valid? That makes no sense...
Exactly that is why is it important for p to be prime.

For the prime case: can you see how the numbers are being rearranged?
6. no... i don't see why the numbers would be just rearrangements, even though every prime i've checked that appears to be the case. How do I show it applies to all primes?
7. bump
8. (Original post by CancerousProblem)
bump
Another hint: use the fact that mod p prime is unique (i.e. no two inverses are the same) for
9. (Original post by tombayes)
(i.e. no two inverses are the same) [/tex]
that's exactly what my friend told me but i have no idea why its true. I can see why if its true it will imply my original question, but i have no idea why it's true. Can you explain why it's true to me? This problem has been disturbing me all day. I wanted to do some C4 past papers today but I came across this in a book and it's been bugging me for 2 hours now before i gave up.
10. (Original post by CancerousProblem)
that's exactly what my friend told me but i have no idea why its true. I can see why if its true it will imply my original question, but i have no idea why it's true. Can you explain why it's true to me? This problem has been disturbing me all day. I wanted to do some C4 past papers today but I came across this in a book and it's been bugging me for 2 hours now before i gave up.
Pick a p prime and
Suppose there exists a positive integer with a non-unique inverse mod p i.e. there exists two integers such that & mod. Then mod this implies or however we already said that was a positive integer so therefore i.e. inverses are unique.

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