Let n be a positive integer. A primitive root mod n is an integer g such that every integer relatively prime to n is congruent to a power of g mod n. And in dealing with modular arithmetic, cyclic groups, and primitive roots, some clear patterns emerge. For example, the number 3 is a primitive root modulo 7, and we can see its cycling here:

We can observe this property with integers in general when dealing with powers. And use a neat trick to deduce the last digit of a number. For example, consider the number 7. As we increment the powers of 7, the last digit begins to repeat: 7, 9, 3, 1, and so on:

7^2 ≡ 49

7^3 ≡ 343

7^4 ≡ 2401

7^5 ≡ 16807

7^6 ≡ 117649

7^7 ≡ 823543

7^8 ≡ 5764801

7^9 ≡ 40353607

How can we guess the last digit of an extremely large number, like \(7^{3001}\)? As we demonstrated, the last digit repeats every 4 numbers. First we divide our exponent, 3001, by 4.

Since we are left with a remainder, we use it, and calculate \(7^{1}\), which is simply 7. Thus, we know the last digit of the number \(7^{3001}\) is 7.

Now, what if our number didn't have a remainder? For example, what if we were calculating \(7^{3000}\)? We would take the same steps and divide the exponent by 4, and have 750 without a remainder, which is equivalent to:

And we would then alternatively calculate \(7^{4}\) which is 240**1**. Thus, the last digit of \(7^{3000}\) is 1.

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