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Convert ${\left( {169} \right)_{10}}$ in base $7$?

Last updated date: 25th Mar 2023
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Hint: The given problem is based on the conversion of the base. In this problem we have to convert a number from base $10$ to base $7$.
For this conversion we divide the base $10$ number by $7$ and if it gives a remainder then we divide its quotient again by $7$.
We repeat this division until Quotient becomes $0$. When the quotient becomes $0$ then we use the digits of remainders from bottom to top to form a base $7$ number. This is the required number.

Complete step by step solution:
We are given the decimal number ${\left( {169} \right)_{10}}$ in base $7$.
We have to convert this number into a base $7$ number.
For this conversion we divide $169$ by $7$, we get
$169 \div 7 = Q\left( {24} \right) + R\left( 1 \right)$
This division gives quotient $\left( Q \right)$ $24$ and remainder $\left( R \right)$$1 Now we divide 24by7, we get 24 \div 7 = Q\left( 3 \right) + R\left( 3 \right) This division gives quotient \left( Q \right)$$3$ and remainder $\left( R \right)$$3, then we divide 3 by 7, we get 3 \div 7 = Q\left( 0 \right) + R\left( 3 \right) This division gives quotient \left( Q \right)$$0$ and remainder $\left( R \right)$$3 Here the quotient becomes 0 so we stop this division. Since the remainders will be the digits of base 7 number. Then, we use the digits of remainders from bottom to top to form a base 7 number. Hence the base 7 number is {\left( {331} \right)_7} Therefore, {\left( {169} \right)_{10}}$$ = {\left( {331} \right)_7}$.

Note:
The number written in base $10$ is called a decimal system and the digits of base $10$ numbers are $0,1,2,3,4,5,6,7,8$ and $9$. Whereas the number written in base is $7$.
The digits of base $7$ numbers are $0,1,2,3,4,5$ and $6$. It should be remembered that to convert a number from any base to any other base it is first converted into base $10$ using expansion method and then it is converted from base $10$ to required base using division and multiplication method.