Simple. (Problem $2$ )

Quick, non-calculator solution: the answer is a positive integer (from the question and from the format of the answer).

It's certainly less than $10$ since $10^9>117649$ .

It's odd, and it's not $5$ (since all powers of $5$ end in the digit $5$ ).

Now, either note that $9^8=81^4$ ends in a $1$ , or check that $1177649$ isn't a multiple of $3$ and the only possibility is $\boxed7$ (which is easily double checked).

• Factorise:

$117649 = \underbrace{7 \times 7 \times 7 \times 7 \times 7 \times 7}_{6 \text{ times}} = 7^{6}$

• Compare:

$\boxed{x = 7}$

$\frac{117649}{7}$ $= 16807$

$\frac{16807}{7}$ $= 2401$

$\frac{2401}{7}$ $= 343$

$\frac{343}{7}$ $= 49$

$\frac{49}{7}$ $= 7$

$\frac{7}{7}$ $= 1$

Since $117649$ gets divided by $7$ exactly $6$ times, $7^6 = 117649$ .

Therefore, $x \rightarrow \fbox{7}$ .