Monoxdifly
MHB
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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?
$$7^1-2^1=7-2=5$$ (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?