Big number

Number Theory Level 3

9 9 9 ⋅ ⋅ ⋅ 9 ⏟ Ninety nine 9’s \Large \underbrace{9^{9^{9^{\cdot^{\cdot^{\cdot{9}}}}}}}_{\text{Ninety nine 9's}}

What is the last digit of the above number?

3 9 1 4 5 8 7 2

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Shithil Islam by shithil Islam , 18, Bangladesh

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2 solutions

Chew-Seong Cheong
Aug 14, 2016

9 9 9 . . . 9 ≡ ( 10 − 1 ) 9 9 . . . 9 (mod 10) ≡ ( − 1 ) 9 9 . . . 9 (mod 10) Since the power is odd. ≡ − 1 (mod 10) ≡ 9 (mod 10) \begin{aligned} 9^{9^{9^{.^{.^{.^9}}}}} & \equiv (10-1)^{9^{9^{.^{.^{.^9}}}}} \text{ (mod 10)} \\ & \equiv (-1)^\color{#3D99F6}{{9^{9^{.^{.^{.^9}}}}}} \text{ (mod 10)} & \small \color{#3D99F6}{\text{Since the power is odd.}} \\ & \equiv -1 \text{ (mod 10)} \\ & \equiv \boxed{9} \text{ (mod 10)} \end{aligned}

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Shithil Islam
Aug 13, 2016

9^2=81,9^3=729. That's means if 9's power is even then it's last disit is 1 and if it's power is odd then the last disit is 9. So the answer is 9.

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