Question

Convert \[{\left( {7396} \right)_{10}}\] to base 5.
A.\[{\left( {241014} \right)_5}\]
B.\[{\left( {214041} \right)_5}\]
C.\[{\left( {142032} \right)_5}\]
D.\[{\left( {211401} \right)_5}\]

Answer 1

Hint: We will perform an iterative procedure of dividing 7396 by 5 and storing the remainder and the quotient. We will proceed until a specific condition is met. We will take all the reminders that we have obtained and arrange them to find the required number.

Complete step-by-step answer:
\[{\left( {7396} \right)_{10}}\] is read as 7396 to the base 10.
To convert this number to base 5, we will need to perform an iterative step; that is, we will repeat the procedure again and again until we meet a specific condition.
The procedure is that we will need to divide the number by 5 and store the remainder and the quotient. The specific condition is that the quotient should be less than 5.
So, we will divide the number by 5 and store the remainder and the quotient. We will again divide the quotient that we have obtained by 5 and store the remainder and the quotient. We will repeat this procedure until we obtain a quotient that cannot be divided by 5 any further; that is, we obtain a quotient that is smaller than 5.
We will divide 7396 by 5. We will store the quotient below the number and the remainder on the right side of the quotient:
\[\begin{array}{l}\left. 5 \right|7396\\\left. {} \right|\left. {1479} \right|1\end{array}\]
We will divide 1479 by 5:
\[\begin{array}{l}\left. 5 \right|\left. {1479} \right|1\\\left. {} \right|\left. {295{\rm{ }}} \right|4\end{array}\]
We will divide 295 by 5:
\[\begin{array}{l}\left. 5 \right|\left. {295{\rm{ }}} \right|4\\\left. {} \right|\left. {59{\rm{ }}} \right|0\end{array}\]
We will divide 59 by 5:
\[\begin{array}{l}\left. 5 \right|\left. {59{\rm{ }}} \right|0\\\left. {} \right|\left. {11{\rm{ }}} \right|4\end{array}\]
We will divide 11 by 5:
\[\begin{array}{l}\left. 5 \right|\left. {11{\rm{ }}} \right|4\\\left. {} \right|\left. {2{\rm{ }}} \right|1\end{array}\]
2 is smaller than 5, we cannot divide 2 by 5, so we will stop the procedure here.
We will write all the steps performed above in the form of an integrated table:
\[\begin{array}{l}\left. 5 \right|\left. {7396} \right|{\rm{ }}\\\left. 5 \right|\left. {1479} \right|1{\rm{ }} \uparrow \\\left. 5 \right|\left. {295{\rm{ }}} \right|4{\rm{ }} \uparrow \\\left. 5 \right|\left. {59{\rm{ }}} \right|0{\rm{ }} \uparrow \\\left. 5 \right|\left. {11{\rm{ }}} \right|{\rm{4 }} \uparrow \\\left. {} \right|\left. {2{\rm{ }}} \right|1{\rm{ }} \uparrow \\{\rm{ }} \to \to \to \end{array}\]
Now, to convert \[{\left( {7396} \right)_{10}}\] to base 5, we will follow the path of the arrows starting from 2 till 1.
So,
 \[ \Rightarrow {\left( {7396} \right)_{10}} = {\left( {21404} \right)_5}\]
\[\therefore \] Option B is the correct option.

Note: The procedure to convert a number to the base of any other number is the same. If we need to convert a number to base 6, we will have to perform the same procedure except that we will divide the given number repeatedly by 6 instead of 5.