maths

3^{56} = 523 347 633 027 360 537 213 511 521

523 347 633 027 360 537 213 511 521 / 7 = 74 763 947 575 337 219 601 930 217 with remainder 2

So find the smallest positive integer b which satisfies 3^56=b(mod7)

b=2

I don't think there can be any other answer, assuming I copied the number down right and did the division correctly (I did it by hand).
I had help from another forum just to get 3^56. My calculator couldn't handle it.
There might be another way to do this but I don't know it. If I get any other info I will pass it on to you.

okay, I've got a much better answer

find the smallest positive integer b which satisfies 3^56=b(mod7)

3^56 = 3^(54+2) = 3^54 * 3^2 = (3^6)^9 * 9

3^6 = 729
729/7=104 remainder 1 = 1 mod 7
9 = 2 mod 7
1 mod 7 * 2 mod 7 = 2 mod 7

b=2