Question

I'm not sure how to start part(a). I believe part (b) is correct, but just in case I would like to make sure it's correct.

(a) Explain p^(q-1) = 1 modq , If p and
q are distinct primes.
Ans: I was thinking of doing
something like part (b) but I
felt it was somehow wrong
because I wouldn't know how to
go from there.

(b) Determine 17^(98) mod 7 (give
answer in mod 7)

Ans: a=17, p=7 a^(p-1)=1 mod p
==> 17^6 = 1 mod 7
[17]^6 = [1]

[17]^98= [(17)^6]^(16) * (17)^2
\__________/
|
[1]^16 * [17]^2 = 289
Therefore, 17^98 = 289 mod7

Answer 1

Part a:

(all expressions are Mod q)

Consider the q-1 numbers p, 2p, 3p, 4p, 5p,...,(q-1)p.

All these numbers are different and nonzero. If ap = bp and p and q don't have divisors in common then a = b. So, a - b must be zero or a multiple of q, which means that all the q - 1 multiples of p are different and nonzero.

Since there are only q-1 nonzero numbers Mod q:
1, 2, 3, ... q-1,

this means that the numbers

p, 2p, 3p, 4p, 5p,...,(q-1)p

are just the numbers

1, 2, 3,... q-1

but in some different order.

This means that:

1*2*3*4*...*(q-1) =

p*(2p)*(3p)*(4p)*...*(q-1)p.

We can write:

p*(2p)*(3p)*(4p)*...*(q-1)p =

p^(q-1) *1*2*3*...*(q-1)

And it follows that:

p^(q-1) = 1



I b)

You can simplify a but more by using that 17 = 3

So, you get 3^2 = 9 = 2 as the answer.