The last digit of a giant power of 3

Question

The number 3^{2026} has hundreds of digits. But you can find its last digit with a tiny bit of pattern-spotting. What is the final (ones) digit of 3^{2026}?

Accepted Answer

The last digit is 9. The last digit of powers of 3 cycles. Only the last digit of each factor affects the last digit of a product, so watch how the ones digit behaves as the exponent climbs: - 3^1 = 3 → ends in 3 - 3^2 = 9 → ends in 9 - 3^3 = 27 → ends in 7 - 3^4 = 81 → ends in 1 - 3^5 = 243 → ends in 3 (the cycle restarts) So the last digit repeats with period 4: 3, 9, 7, 1, 3, 9, 7, 1, \dots Find the exponent's place in the cycle. Divide the exponent by 4 and take the remainder: 2026 = 4 	imes 506 + 2 \quad\Rightarrow\quad 2026 \bmod 4 = 2. A remainder of 2 points to the second entry in th…

The last digit of a giant power of 3

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