Challenge

Number Theory Level 1

What are the last digit of 3 to the power of 100?

Hint: x^6 * x^4= x^10


The answer is 1.

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Ahmet Ozer by Ahmet Ozer , 20, USA

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3 solutions

We're looking for 3 100 ( m o d 10 ) 3^{100}\pmod{10} Since φ ( 10 ) = 4 \varphi(10)=4 , then the equation becomes: 3 100 − 25 × 4 ( m o d 10 ) 3^{100-25\times4}\pmod{10} 3 0 ( m o d 10 ) 3^0\pmod{10} 1 ( m o d 10 ) 1\pmod{10}

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Ahmet Ozer
Dec 3, 2014

3^1=3

3^2=9

3^3=27

3^4=81

3^5=243

There is a pattern above of the last digit^^^^^

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Parveen Soni
Dec 3, 2014

3^100=(3^4)^25 and 3^4 always has unit digit as 1. Sp 1^25 =1

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