The tetragun!

Number Theory Level 4

Tetration is defined as

n a = a a ⋅ ⋅ a ⏟ n \large {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n

Calculate the value of 7 7 ( m o d 100 ) ^{7}7 \pmod {100} .

Image credit: Tetration.org


The answer is 43.

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Rohit Udaiwal by Rohit Udaiwal , 20, India

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1 solution

Otto Bretscher
Nov 10, 2015

Nice problem! The numbers work out very well...

We compute 7 7 ≡ 7 3 ≡ 43 ( m o d 100 ) 7^7\equiv 7^3\equiv 43\pmod{100} . Now 7 7 7 ≡ 7 43 ≡ 7 3 ≡ 43 ( m o d 100 ) 7^{7^7}\equiv 7^{43}\equiv 7^3 \equiv 43\pmod{100} . In the exponent we can work modulo 20 = λ ( 100 ) 20=\lambda(100) , using the Carmichael lambda.

The sequence becomes stationary after one step, at 43 \boxed{43}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

I travelled up the whole ladder of 7's using the Euler totient function D:

Jake Lai - 5 years, 7 months ago

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