The last digit of seven to the seven to the seven

Question

What is the last digit of 7 to the power of (7 to the power 7)? That is, the exponent is 7 to the 7 = 823,543, and the number itself has about 700,000 digits. Explain the method so that it works for any tower of exponents, and find the last two digits as well.

Accepted Answer

3. The cycle. Last digits of powers of 7: 7, 49 (9), 343 (3), 2401 (1), then 7 again. The pattern 7, 9, 3, 1 repeats with period 4. So the last digit of 7 to the n depends only on n modulo 4: remainder 1 gives 7, remainder 2 gives 9, remainder 3 gives 3, remainder 0 gives 1. The exponent modulo 4. The exponent is 7 to the 7. Since 7 leaves remainder 3 when divided by 4, and 3 squared = 9 leaves remainder 1, powers of 7 modulo 4 alternate 3, 1, 3, 1, ... Odd powers leave remainder 3. So 7 to the 7 leaves remainder 3 modulo 4. (Directly: 7 to the 7 = 823,543, and 823,543 = 4 × 205,885 + 3.) Con…

The last digit of seven to the seven to the seven

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