Question

Find the value of x such that 365 based seven + 43 based x = 217 based 10.

Answer 1

We need to find the base x in the following equation:

$365_7+43_x=217_{10}$

First, lets convert 365 from base 7 to base 10. This is given by

$365_7=3\times7^2+6\times7^1+5\times7^0$

where the upperindex denotes the position of eah number. This gives

$\begin{gathered} 365_7=3\times49+6\times7+5\times1 \\ 365_7=147+42+5 \\ 365_7=194_{10} \end{gathered}$

that is, 365 based 7 is equal to 194 bases 10.

Now, lets do the same for 43 based x. Lets convert 43 based x to base 10:

$43_x=4\times x^1+3\times x^0$

where again, the superindex 0 and 1 denote the position 0 and 1 in the number 43. This gives

$43_x=(4x+3)_{10}$

Now, we have all number in base 10. Then, our first equation can be written in base 10 as

$194_{10}+(4x+3)_{10}=217_{10}$

For simplicity, we can omit the 10 and get

$194+4x+3=217$

so, we can solve this equation for x. By combining similar terms. we have

$197+4x=217$

and by moving 197 to the right hand side, we obtain

$\begin{gathered} 4x=217-197 \\ 4x=20 \end{gathered}$

Finally, we get

$\begin{gathered} x=\frac{20}{4} \\ x=5 \end{gathered}$

Therefore, the solution is x=5