Can You Prove: 𝟑^𝒏+𝟏 Is Not Divisible by 𝟖 | Olympiad Math

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For even and odd powers, we use the binomial theorem to complete the proof.

  1. Dr. Wang
  2. use the binomial theorem to complete the proof.
  3. Dr. Wang
  4. Olympiad Math
  5. 𝟑^𝒏 𝟏 is not divisible by 𝟖
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Комментарии
Автор

We can use modular arithmetic directly. 3^n is congruent to either 1 or 3 mod (8). Thus 3^n + 1 shall be congruent to 2 or 4 mod(8) which is not divisible by 8. Hence proved.

amitsrivastava
Автор

Let m=n-1. 3^n+1=3^(m+1)+1. For m=0 (or n=1) it is divisible by 4. For m>0 (or n>1) it is not divisible by 4 as 3^(m+1)+1=3^m+4 which is not ivisible by 4 since 3^m is not divisible by 4. A number is divisible by 8 means that the number is also divisible by 4. As 3^n+1=3^(m+1)+1 which is not divisible by 4 automatically it is not divisible by 8. Eventhough 3^n+1 is divisible by 4 it is not divisible by 8. Hence 3^n+1 is not divisible by 8 for any value of n.
Is my argument correct Sir?

nasrullahhusnan
Автор

Although this is correct, I think contradiction proof will be a better one. Please do not assume some facts and skip them.

Biologymus