Question

If the decimal number ${{2}^{111}}$ is written in the octal system, then what is the digit in its unit place.

Answer 1

Hint: To convert the given decimal number into octal system, we first need to divide the given number until the quotient is less than 8. Similarly, we divide ${{2}^{111}}$ with 8 and find the value of quotient and remainder. We use the fact that obtained remainders are written last in first order and find the digit in its unit place of octal system.

Complete step by step answer:
Given that we have a decimal number ${{2}^{111}}$ and we need to write the given number in the octal system.
We need to determine what is the digit in the unit place when the given number is converted into the octal system.
We know that for converting a given decimal to octal, we need to follow the series of steps as shown below:
i) We check whether the given decimal number is less than 8, if yes the same will be the number in the octal system.
ii) If not, we divide the given decimal number with 8 and note down the obtained quotient and remainder.
iii) Now, we observe that the obtained remainder will be the octal number in its unit place.
iv) If the obtained quotient is greater than or equal to 8 divide it again with 8 until we get a quotient less than 8.
v) We note the remainder whenever we divide the quotient.
vi) Now, we write the obtained remainders in the last in order to get the octal system number.
Now we have a decimal number ${{2}^{111}}$ and we convert it into the powers of 8.
We know that ${{2}^{3}}=8$, using these we convert ${{2}^{111}}$.
${{2}^{111}}={{\left( {{2}^{3}} \right)}^{\dfrac{111}{3}}}$.
${{2}^{111}}={{8}^{37}}$.
We can see that ${{8}^{37}}$ is clearly divisible by 8 and we get remainder to be 0.
So, if we convert ${{2}^{111}}$ into the octal system we get 0 in its unit place as the first obtained remainder will be the number in units place.

∴ The digit in unit place after converting ${{2}^{111}}$ to octal system is 0.

Note: Here we have converted the given decimal number ${{2}^{111}}$ to ${{8}^{37}}$ for reducing the calculation time to get remainder. If we need digits in places other than unit place we divide until the remainder of the required place appears. Similar type of technique can be used to decimal to binary by dividing given decimal number with 2 and decimal to hexa-decimal by dividing given decimal with 16.