Smallest Prime Number

Algebra Level 2

What is the smallest prime factor of 2 111 + 3 111 ? \large 2^{111}+3^{111}?

2 2 2 111 2^{111} None of the above 3 111 3^{111} 3 3

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Anirudh Sreekumar
Oct 14, 2017

2 111 + 3 111 ( m o d 2 ) = 0 111 + 1 111 ( m o d 2 ) = 1 ( m o d 2 ) 2 111 + 3 111 ( m o d 3 ) = ( − 1 ) 111 + 0 111 ( m o d 3 ) = − 1 ( m o d 3 ) ⟹ It is not divisible by 2 or 3. \begin{aligned}2^{111}+3^{111} \pmod{2}&=0^{111}+1^{111} \pmod{2}=1 \pmod{2}\\ 2^{111}+3^{111} \pmod{3}&=(-1)^{111}+0^{111} \pmod{3}=-1 \pmod{3}\\\\ \implies\text{It is not divisible} &\text{ by 2 or 3.}\end{aligned}

Note that, \text{Note that,}

2 ( m o d 5 ) = − 3 ( m o d 5 ) 2 111 + 3 111 ( m o d 5 ) = ( − 3 ) 111 + 3 111 ( m o d 5 ) = 0 ( m o d 5 ) Thus 5 is the smallest prime that divides 2 111 + 3 111 \begin{aligned}2\pmod{5}&=-3\pmod{5}\\ 2^{111}+3^{111} \pmod{5}&=(-3)^{111}+3^{111} \pmod{5}=0 \pmod{5}\\\\ \text{Thus 5 is the smallest} &\text{ prime that divides } 2^{111}+3^{111}\end{aligned}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

That is the right solution. Also, we can generalize it to 2 n + 3 n 2^n+3^n . Thank you for sharing your solution.

Hana Wehbi - 3 years, 7 months ago

option should be none of the below :)

Harinder Choudhary - 3 years, 7 months ago

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