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This question is asked before here, but an easily grasped answer is not given (Without modular arithmetic). I'm facing the same doubt that this friend faced in $2017$:

I'll state the question here:

If "$P(n):49^n+16^n+k$ is divisible by $64$ for all $n∈N$" is true, then what is the least negative integral value of k?

The books says it is $k=-1$. But this ain't true for all $n∈N$. It's only true for $n=1$.

The previous answers on this site say no such $k$ exists. Can you explain this without using modular arithmetic.

Simran
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    But modular arithmetic is precisely the right tool for this sort of problem. – Angina Seng Sep 06 '19 at 00:51
  • This is a grade $11$ question is India. It's in our syllabus but we haven't done modular arithmetic yet. – Simran Sep 06 '19 at 00:53
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    We can easily hide the modular arithmetic: Divide $k$ by $64$ by the euclidean algorithm: $k=64p+r$ for some $0\leq r\leq 63$. Then divide $49+16+k$ and $49^2+16^2+k$ by $64$. Compare the remainders and get to a contradiction. – Luiz Cordeiro Sep 06 '19 at 00:55

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Hint: If both $49^1+16^1+k$ and $49^2+16^2+k$ are divisible by $64$, so is their difference.

Another hint: Apparently, your book is wrong.

Luiz Cordeiro
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