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It is a 3.

Look at the ones digit of successive powers of 7; this need only be done by considering the multiplication of the ones digit of the previous power of 7 by 7 (as this is the only calculation that affects the ones digit as each successive power of 7 is the previous power multiplied by 7) and taking the result modulus 10 (to extract the new ones digit as any excess over 9 is carried into the tens column):

7¹ → 7 mod 10 = 7

7² → (7×7) mod 10 = 9

7³ → (9×7) mod 10 = 3

7⁴ → (3×7) mod 10 = 1

7⁵ → (1×7) mod 10 = 7

At this point the pattern of the ones digit will obviously repeat the sequence of the four digits {7, 9, 3, 1}.

To find the ones digit of any power of 7, take that power modulus 4 use that digit from the four digit sequence. Note that when taking the number modulus 4, the result will be in the range 0-3; when the result is 0, use the 4th digit from the sequence.

2015 mod 4 = 3

→ the third digit of {7, 9, 3, 1}, which is 3, will be the ones digit of 7²⁰¹⁵.

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