Hoe to solve this?We have hat3 inZZ_7;Calculate hat3^6andhat3^2014

1 Answer
George C.
Apr 6, 2017

hat(3)^6 = hat(1)

hat(3)^2014 = hat(4)

Explanation:

Writing hat(a) for the equivalence class of a modulo 7, we find:

hat(3)^0 = hat(1)

hat(3)^1 = hat(3)

hat(3)^2 = hat(3*3) = hat(9) = hat(2)

hat(3)^3 = hat(3*2) = hat(6)

hat(3)^4 = hat(3*6) = hat(18) = hat(4)

hat(3)^5 = hat(3*4) = hat(12) = hat(5)

hat(3)^6 = hat(3*5) = hat(15) = hat(1)

We could have deduced hat(3)^6 = hat(1) from Fermat's little theorem, which can be stated:

If p is a prime number, then for any integer a:

a^p -= a" " modulo p

If in addition a != 1 modulo p, then we can divide by a to find:

a^(p-1) -= 1" " modulo p

From hat(3)^6 = hat(1), we can deduce:

hat(3)^(6m+n) = hat(3)^n

So:

hat(3)^2014 = hat(3)^(6*335+4) = hat(3)^4 = hat(4)

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