Why only odd numbers?

Number Theory Level 4

172 9 172 7 172 5 172 3 ⋅ ⋅ ⋅ 3 1 \Large 1729^{ 1727^{ 1725^{ 1723^{ \cdot^{ \cdot^{\cdot^{3^{1}}}}}}}}

Find the last 2 digits of the above number.


The answer is 9.

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Akshat Sharda by Akshat Sharda , 21, India

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1 solution

Otto Bretscher
Mar 22, 2016

Since λ ( 100 ) = 20 \lambda(100)=20 and λ ( 20 ) = 4 \lambda(20)=4 , where λ \lambda denoted the Carmichael's lambda function , the given number is congruent to 2 9 7 1 = ( − 1 + 30 ) 7 ≡ − 1 + 7 × 30 ≡ 9 ( m o d 100 ) 29^{7^1}=(-1+30)^7\equiv -1+7\times 30\equiv \boxed{9} \pmod{100}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice solution. The Carmichael Lamba really does make calculations easier.

Abdur Rehman Zahid - 5 years, 2 months ago

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