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*7^2+6*7^1+5*7^0`

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

`\begin{gathered} 365_7=3*49+6*7+5*1 \\ 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* x^1+3* 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=(20)/(4) \\ x=5 \end{gathered}`

Therefore, the solution is x=5