Find the unit digit of...

Number Theory Level 2

2 7 3 4 4 3 29 \Large 27^{34^{43^{29}}}

Find the unit digit of the number above.

7 3 1 9

Bulbuul Dev by Bulbuul Dev , 37, India

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1 solution

Chew-Seong Cheong
Apr 18, 2018

2 7 3 4 4 3 29 ≡ ( 20 + 7 ) 3 4 4 3 29 (mod 10) ≡ 7 3 4 4 3 29 (mod 10) ≡ 4 9 2 4 3 29 − 1 × 1 7 4 3 29 (mod 10) ≡ ( 50 − 1 ) 2 4 3 29 − 1 × 1 7 4 3 29 (mod 10) ≡ 1 (mod 10) \large \begin{aligned} 27^{34^{43^{29}}} & \equiv (20+7)^{34^{43^{29}}} \text{ (mod 10)} \\ & \equiv 7^{34^{43^{29}}} \text{ (mod 10)} \\ & \equiv 49^{2^{43^{29}-1} \times 17^{43^{29}}} \text{ (mod 10)} \\ & \equiv (50-1)^{2^{43^{29}-1} \times 17^{43^{29}}} \text{ (mod 10)} \\ & \equiv \boxed{1} \text{ (mod 10)} \end{aligned}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Isn't it simpler to realise a power of 34 must be a multiple of 4 as there will be multiple 2s in the prime factoring. For 4n powers of 7, the unit digit is 1.

Theodore Sinclair - 3 years, 1 month ago

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You are right.

Chew-Seong Cheong - 3 years, 1 month ago

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