GATE CSE 1991 | Question: 01-iii

Question

Consider the number given by the decimal expression:
$$16^3*9 + 16^2*7 + 16*5+3$$
The number of $1’s$ in the unsigned binary representation of the number is ______

Comments

$16^3=2^{12}$ and multiplying a number in binary with $2^x$ means that we are shifting that number to the left by $x$ bits.

So, the number $16^3 . 9$ will be $1001(=9)$ shifted by $12$ bits to the left which will become $1001000000000000$.

Same we can do with other numbers as well and will find that there is no overlapping $1's$. So total $1's = 1's$ in $9,7,16,3$.

So the answer is $9$.

Result is $1001011101010011$.

Answer 1

Convert this number into hexadecimal first, So to do this I will divide the given number by 16 

$16^{3}*9+16^{2}*7+16*5+3$ 

as we can see that 16 is a common term in all except the last term, so on dividing by 16 we get 3 as remainder so next time the expression would be $16^{2}*9+16*7+5$ again upon division by 16 we would get 5 as remainder. Similarly, we would keep dividing and we would get 7 and 9 as the remainder respectively.

So the number in hexadecimal would be 9753 and we would just convert it into binary we get $(1001 0111 0101 0011)_{2}$

Hence 9 1's are there.