Find the remainder

Number Theory Level 1

What's the remainder, if we divide 2 2014 + 5 2014 { 2 }^{ 2014 }+{ 5 }^{ 2014 } by 3 3 ??


The answer is 2.

Md Mehedi Hasan by Md Mehedi Hasan , 22, Bangladesh

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1 solution

2 2014 + 5 2014 = ( 3 − 1 ) 2014 + ( 2 × 3 − 1 ) 2014 ≡ ( ( − 1 ) 2014 + ( − 1 ) 2014 ) ( m o d 3 ) ≡ ( 1 + 1 ) ( m o d 3 ) ≡ 2 ( m o d 3 ) 2^{2014} + 5^{2014} = (3 - 1)^{2014} + (2 \times 3 - 1)^{2014} \equiv ((-1)^{2014} + (-1)^{2014}) \pmod{3} \equiv (1 + 1) \pmod{3} \equiv \boxed{2} \pmod{3} .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Just as a side note, entering your answer is probably not the best answering method for this, as you get 3 tries and there are only three answers = 0, 1 and 2. So, by just trying those three, anyone can get it right

Adrian Ma - 1 year ago

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