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.
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.