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

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.