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
- 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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