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1. The problem statement, all variables and given/known data

Prove by mathematical induction that for all +ve integers n,[itex]10^{3n}+13^{n+1}[/itex] is divisible by 7.

2. Relevant equations

3. The attempt at a solution

Assume true for n=N.

[tex]10^{3N}+13^{N+1}=7A[/tex]

Multiply both sides by ([itex]10^3+13[/itex])

[tex](10^{3N}+13^{N+1})(10^3+13)=7A(10^3+13)[/tex]

[tex]10^{3N+3}+ 10^3(13^{N+1})+13(10^{3N})+13^{N+2}=7A(1013)[/tex]

[tex]10^{3N+3}+13^{N+2}=7A(1013)-10^3(13^{N+1})-13(10^{3N})[/tex]

Here is where I am stuck. I need to show that [itex]10^3(13^{N+1})-13(10^{3N})[/itex] is divisible by 7 now.

What I would like to get is that [itex]10^3(13^{N+1})-13(10^{3N})[/itex] can somehow be manipulated into the initial inductive hypothesis and then it will become true for n=N+1. So I need some help.

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# Proof by mathematical induction

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