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Number Theory Level 2

What is the rightmost digit of 7 2014 7^{2014} ?

None of these choices 9 5 3 1 7

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Jack Sacks by Jack Sacks , 18, United Kingdom

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2 solutions

Arulx Z
Feb 14, 2016

7 2014 ≡ 49 1007 ≡ ( − 1 ) 1007 ≡ − 1 ≡ 9 ( mod 10 ) { 7 }^{ 2014 }\equiv { 49 }^{ 1007 }\equiv { \left( -1 \right) }^{ 1007 }\equiv -1\equiv 9\quad \left( \text{mod 10} \right)

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Moderator note:

Simple standard approach.

Harsh Khatri
Feb 14, 2016

7 2014 = 7 2 ⋅ 7 503 × 4 = 49 ⋅ 7 503 × ϕ ( 10 ) = 49 ⋅ 1 ≡ 9 ( m o d 10 ) 7^{2014} = 7^{2}\cdot 7^{503\times4} = 49 \cdot 7^{503\times \phi(10)} = 49\cdot 1 \equiv 9 \pmod{10}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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