Solve without a calculator – 4

Since the last digit is 7, we can immediately reject the possibilities of numbers ending with 3 or 5 because their 5th powers don't have unit digit 7.

Now we are only left with 27,37 and 47.

Check divisibility by 3...Clearly it's not divisible.Since it's not divisible by 3, there's no point for it to be divisible by 27.

So we are left with 37 or 47.

Now we can easily calculate $40^5$ which is nothing but $4^5\times 10^5$ which is 102400000.... Now what can we conclude from here?

It has 9 digits... and the number given is of 8 digits... So answer has to be $\boxed{37}$

Check this from Numberphile.

Clearly the number ends with 7,

and it lies between (30)^7 and (40)^7

Therfore 37