Consider the first 8 calculations of 7 to an exponent:

7, 9, 3, 1, 7, 9, 3, 1

The 7, 9, 3, 1 repeats through infinity.

So every factor of 4, the cycle of 7, 9, 3, 1 restarts.

Counting backwards from 2013, we know that 2012 is the largest number divisible by 4:

7^2013 = 7^2012 * 7^1

The cycle starts over after 2012.

Which means the last digit of 7^2013 =

- 7^1 = 7
- 7^2 = 49
- 7^3 = 343
- 7^4 = 2,401
- 7^5 = 16,807
- 7^6 = 117,649
- 7^7 = 823,543
- 7^8 = 5,764,801

7, 9, 3, 1, 7, 9, 3, 1

The 7, 9, 3, 1 repeats through infinity.

So every factor of 4, the cycle of 7, 9, 3, 1 restarts.

Counting backwards from 2013, we know that 2012 is the largest number divisible by 4:

7^2013 = 7^2012 * 7^1

The cycle starts over after 2012.

Which means the last digit of 7^2013 =

**7**

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