Question

In the familiar decimal number system the base is 10. In another number system using base 4 the counting proceeds as 1,2,3,10,11,12,13,20,21,..... The twentieth number in this system will be
A. \[40\]
B. \[320\]
C. \[210\]
D. \[110\]

Answer 1

Hint: Using the knowledge of a quaternary (base \[4\]) number system write the numbers in ascending order and count the twentieth term from left side.
A quaternary (base \[4\]) number system uses four digits \[0,1,2,3\] to write any number.
* Number system with base \[10\] uses digits \[0,1,2,3,4,5,6,7,8,9\]
We write numbers in ascending order as \[1,2,3,4,5,6,7,8,9,10,11,12,1,3,14,15.....20,21,22,...\].

Complete step by step answer:
In a number system with base \[4\] we only use digits \[0,1,2,3\]
Therefore, we write
\[1,2,3\] and then the fourth digit becomes \[10\].
Next terms go the same way till \[13\] and then we write \[20\] and write terms till \[23\].
Moving in the same manner write the terms,
\[1,2,3,10,11,12,13,20,21,22,23,30,31,32,33,100,101,102,103,110,111,112....\]
Now counting from the left side, we can check the twentieth term comes out to be \[110\].

Therefore, option D is correct.

Note:
Students are likely to make mistakes in counting when they don’t know that the base \[4\] number system does not contain the digit \[4\] and only contains four digits including \[0\] (some students make the mistake of starting the digits from \[1\] ).
Alternative method:
Since, the twentieth term from base \[10\] is \[20\].
We can use the method of conversion of a number from base \[10\] to base \[4\].
This method involves division of numbers from \[4\] multiple times till we obtain both remainder and quotient from the set \[\{ 0,1,2,3\} \].
Therefore, we just keep dividing by \[4\] and use our successive remainder as the next number on the string.
Start by dividing the number by \[4\],
\[
  4\mathop{\left){\vphantom{1{20}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{20}}}}
\limits^{\displaystyle \,\,\, 5} \\
   - 20 \\
  \overline { = 0} \\
 \]
i.e. \[20 \div 4 = 5\]
Remainder is zero so we don’t need to do any more division by four.
So, the remainder here is zero and the quotient is \[5\].
But \[5\] does not exist in base \[4\] number system
Given the terms \[1,2,3,10,11,12,13,20,21,....\]
The fifth term in the base \[10\] number system will be the fifth term in the base \[4\] number system as well.
So, \[5\] from base ten is converted to \[11\] in base four.
Therefore, we can write \[20\] as \[110\] in base four.

So, option D is correct.