Using Mathematical Induction to prove that $\forall n \ge 0$ that $5\mid (8^n−3^n)$

Question

$\forall n \ge 0$ : $5\mid 8^n−3^n$, therefore $8^n − 3^n = 5m$, with $m \in \mathbb{Z}$.

For example $8^2 − 3^2 = 55 = 5 \cdot 11$ and $8^3 − 3^3 = 485 = 5 \cdot 97$.

I know that the first step would be something regarding showing that it is true for the first case, usually $n = 1$. Then for step 2, I would assume that it is true for $n = k$, and prove it is true for $n = k + 1$. I am unsure how to apply it with $\forall n \geq 0$ : $5$?

$$8^{n+1} - 3^{n+1} = 3 \times \left[8^n - 3^n\right] ~\color{red}{+ ~\text{what ?}}$$ — user2661923, Jun 17 '22 at 21:18
...or you could use $$8^n-3^n=(8-3)\left(8^{n-1}+8^{n-2}\cdot 3+\ldots+8\cdot 3^{n-2}+3^{n-1}\right)$$ — DonAntonio, Jun 17 '22 at 21:25
"Then for step 2, I would assume that it is true for n=k, and prove it is true for n=k+1." Did you manage to do this? — DanielV, Jun 17 '22 at 21:57
Does this answer your question? Prove that $5\mid 8^n - 3^n$ for $n \ge 1$ — , Jun 18 '22 at 22:38

score 3 · Accepted Answer · answered Jun 17 '22 at 21:29

3

Here's a sketch of the answer

Step 1:(n = 0)

$$ 8^0 - 3^0 = 0 = 5*0 $$

Step 2: Assuming the propety is valid for n = k, we want to show that it is valid for n = k+1.

$$ 8^{k+1} - 3^{k+1} = 3(8^k - 3^k) + (8^{k+1} - 3*8^k) = 3(8^k - 3^k) + 8^k(8 - 3) = 3(8^k - 3^k) + 5*8^k $$

By our induction hypothesis we have that $(8^k - 3^k) = 5m$:

$$ 3(8^k - 3^k) + 5*8^k = 3*5m+5*8^k = 5(3m+8^k) $$

answered Jun 17 '22 at 21:29

JazzyStudent

116

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This is actually easier to see using mod 5, 8 = 3 (mod 5). So $8^k - 3^k = 3^k - 3^k (mod 5) \Rightarrow 8^k - 3^k = 0 (mod 5)$. – JazzyStudent Jun 17 '22 at 21:34
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This question goes against our quality standard policy. Instead of posting an answer, you should encourage the person who posted the question to improve it. – José Carlos Santos Jun 17 '22 at 21:34
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Isn't your answer a repeat of my comment? – user2661923 Jun 17 '22 at 21:34
Maybe he wanted help on how to write it properly. He clearly knew what he needed to do. – JazzyStudent Jun 17 '22 at 21:38
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@JazzyStudent The assumption that modulus commutes with exponents is exactly equivalent to the question being asked, so it isn't really easier, it's just stating the question in a different language. – DanielV Jun 17 '22 at 21:55
Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Jun 19 '22 at 07:48

Using Mathematical Induction to prove that $\forall n \ge 0$ that $5\mid (8^n−3^n)$

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