Thread: more modulo proofs

any help on getting started with these would be great. I seem to be having a dead brain day.

• Let p be a prime number.Suppose you take all the non zero integers 0,1,2,...,p-1, multiply them together, and take the answer modulo p.Show that you will always get p-1

• Determine which of 1,2,3,4,5,6,7,8,9,10 (the non zero integers modulo 11) are squares modulo 11. Use this to find the roots of x^5 - 1 modulo 11

• Let f(x) be a polynomial with integer coefficients. Show that if the value of f(x) is divisible by r at r consecutive integers, then f(m) is divisible by are for all integers m.

Thank you

Originally Posted by gpenguin

any help on getting started with these would be great. I seem to be having a dead brain day.

• Let p be a prime number.Suppose you take all the non zero integers 0,1,2,...,p-1, multiply them together, and take the answer modulo p.Show that you will always get p-1

• Determine which of 1,2,3,4,5,6,7,8,9,10 (the non zero integers modulo 11) are squares modulo 11. Use this to find the roots of x^5 - 1 modulo 11

• Let f(x) be a polynomial with integer coefficients. Show that if the value of f(x) is divisible by r at r consecutive integers, then f(m) is divisible by are for all integers m.

Thank you

What've you done so far?

Tonio

• let p be a prime number...

for p= 3
2 (mod 3) = 1

which isn't true and then confuses me

• determine which of ...

I don't know what squares modulo 11 means

• let f(x)=...

eg r= 2
if f(m) and f(m+1) are divisible by 2 then f(m) is always divisible by 2

and then i don't know how to compute this
I need to do r=2 and r=3

2(mod 3) = 2.....not 1.

That helps with proving through numbers. But I still can't prove it generally.

Originally Posted by gpenguin

• Determine which of 1,2,3,4,5,6,7,8,9,10 (the non zero integers modulo 11) are squares modulo 11. Use this to find the roots of x^5 - 1 modulo 11

For example, 1*1 = 1 is a perfect square. 2*2 = 4 is a perfect square. Something a little less obvoius this time... 5*5 = 3, so 3 is a perfect square. Just run the list from 1*1 to 10*10 (mod 11). That will generate your list.

-Dan