#### If $6\cdot4=26$, $6\cdot5=33$, $6\cdot6=40$, and $6\cdot7=46$, what does $4\cdot5\cdot6=$?

143

Why? Well if we look at this logically, there is a clear base conversion here. When we have base $N$, we know that the way to convert it back into decimal ($\text{Base}_{10}$). We multiply each digit by the base raised to the degree of one less than that digit's place.

Example: $2012_{3}\to\text{base}_{10}$

$\to 2(3)^{3}+0(3)^{2}+1(3)^{1}+2(3)^{0}$

$\to 2(27)+0(9)+1(3)+2(1)$

$\to 54+0+3+2$

$=59$

So, where in decimal, the answers would be $24,30,36,42$ respectively, we get the sequence $26,33,40,46$. So we work through what we know.

$24_{10}\to 26_{N}$

Well, let's write it out like we did above so that $2(N)^{1}+6(N)^{0}=24$

$\to 2N+6=24$

$\to 2N=18$

$\therefore N=9$.

It is all in base 9. We can use a base-converter like the one below to do this, or we can use our knowledge of base-conversion. For now though, let's just use the tool below.