[ASK] Mathematical Induction: Prove 7^n-2^n is divisible by 5.

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The discussion focuses on proving that \(7^n - 2^n\) is divisible by 5 using mathematical induction. The initial steps confirm the base case for \(n = 1\) and assume the statement holds for \(n = k\). A mistake is identified in the transition to \(n = k + 1\), where the correct expression reveals that \(12 \cdot 2^k\) is not divisible by 5. Instead, a simpler method involving the difference of two terms with the same power is suggested, although participants note that this approach isn't part of the school curriculum. Ultimately, the conversation emphasizes the importance of learning efficient proof techniques.
Monoxdifly
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Prove by mathematical induction that $$7^n-2^n$$ is divisible by 5.What I've done so far:For n = 1

$$7^1-2^1=7-2=5$$ (true that it is divisible by 5)

For n = k

$$7^k-2^k=5a$$ (assumed to be true that it is divisible by 5)

For n = k + 1

$$7^{k+1}-2^{k+1}=7^k\cdot7-2^k\cdot2=7(7^k-2^k)+12\cdot2^k=7(5a)+12\cdot2^k$$

This is where the problem lies. How can I show that $$12\cdot2^k$$ is divisible by 5?
 
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Monoxdifly said:
Prove by mathematical induction that $$7^n-2^n$$ is divisible by 5.What I've done so far:For n = 1

$$7^1-2^1=7-2=5$$ (true that it is divisible by 5)

For n = k

$$7^k-2^k=5a$$ (assumed to be true that it is divisible by 5)

For n = k + 1

$$7^{k+1}-2^{k+1}=7^k\cdot7-2^k\cdot2=7(7^k-2^k)+12\cdot2^k=7(5a)+12\cdot2^k$$

This is where the problem lies. How can I show that $$12\cdot2^k$$ is divisible by 5?
Hi Monoxdifly,

$12\cdot2^k$ is certainly not divisible by $5$ (look at the prime factors).

You made a mistake in your calculation: I get:
$$
7^k\cdot7 - 2^k\cdot2 = 7(7^k-2^k) + 5\cdot2^k
$$
and this should clear things up.
 
Ah, I see. Thank you very much!
 
It seems a bit overkill to use Induction, when there's a simple rule for the Difference of Two Terms With the Same Power...

$\displaystyle \begin{align*} a^n - b^n \equiv \left( a - b \right) \sum_{r = 0}^{n - 1}{ a^{n - 1 - r}\,b^r } \end{align*}$

so in your case the factor would be (7 - 2) which equals 5.
 
Prove It said:
It seems a bit overkill to use Induction, when there's a simple rule for the Difference of Two Terms With the Same Power...

$\displaystyle \begin{align*} a^n - b^n \equiv \left( a - b \right) \sum_{r = 0}^{n - 1}{ a^{n - 1 - r}\,b^r } \end{align*}$

so in your case the factor would be (7 - 2) which equals 5.

Well, the school curriculum doesn't teach this rule, so yes, we were supposed to solve it with the "overkill" method.
 
Monoxdifly said:
Well, the school curriculum doesn't teach this rule, so yes, we were supposed to solve it with the "overkill" method.

Then you can consider it as something new that you have learnt. The most concise method of proof is always the best.
 
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