[ASK] Mathematical Induction

Monoxdifly

Prove by mathematical induction that $$\displaystyle 7^n-2^n$$ is divisible by 5.

What I've done so far:

For n = 1
$$\displaystyle 7^1-2^1=7-2=5$$ (true that it is divisible by 5)
For n = k
$$\displaystyle 7^k-2^k=5a$$ (assumed to be true that it is divisible by 5)
For n = k + 1
$$\displaystyle 7^{k+1}-2^{k+1}=7^k\cdot7-2^k\cdot2=7(7^k-2^k)+12\cdot2^k=7(5a)+12\cdot2^k$$
This is where the problem lies. How can I show that $$\displaystyle 12\cdot2^k$$ is divisible by 5?

Idea

$$\displaystyle 7^{k+1}-2^{k+1}=7\left(7^k-2^k\right)+5\cdot 2^k$$

Monoxdifly

Monoxdifly

Ah, I see. Thank you very much!

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