842 views
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 ______
edited | 842 views

Hex representation of given no. is (9753)16

Its binary representation is (1001 0111 0101 0011)2

The no. of 1's is 9

selected by

$16^3∗9+16^2∗7+16∗5+3$
In Binary representation, each number which can be represented in the power of 2 contains only one 1.
For example 4 = 100, 32 = 100000
$16^3∗9+16^2∗7+16∗5+3$
Convert the given expression in powers of 2.
$16^3∗9+16^2∗7+16∗5+3$
$2^{12}∗(8+1)+2^8∗(4+2+1)+2^4∗(4+1)+(2+1)$
$2^{15}+2^{12}+2^{10}+2^9+2^8+2^6+2^4+2^1+ 2^0$

There are total 9 terms, hence, there will be nine 1's. for example 4+2 = 6 and 4 contains two 1's.

We can solve this also in a different way.
See 2, 4 8, 16..... can be written as
010000...(x times 0 depend upon the number)
i.e
2 = 010
4 = 0100
8 = 01000
16=010000 and so on.
So now if any number Y multiply with any of the above the answer will be Y followed by x times 0.
i.e.
if any number say 7 is multiplied with 16 answer will be
1110000.
So number of 1 will be the number of 1 in that number as multiplication with 2, 4 8 ,6.... can only increase 0 on it.

So  number of 1 in
16^3∗9+16^2∗7+16∗5+3
is equal to
number of 1 in 9+
number of 1 in 7+
number of 1 in 5+
number of 1 in 3
=2+3+2+2
=9
We can solve this also in a different way.
See 2, 4 8, 16..... can be written as
010000...(x times 0 depend upon the number)
i.e
2 = 010
4 = 0100
8 = 01000
16=010000 and so on.
So now if any number Y multiply with any of the above the answer will be Y followed by x times 0.
i.e.
if any number say 7 is multiplied with 16 answer will be
1110000.
So number of 1 will be the number of 1 in that number as multiplication with 2, 4 8 ,6.... can only increase 0 on it.

So  number of 1
is equal to
number of 1 in 9+
number of 1 in 7+
number of 1 in 5+
number of 1 in 3
=2+3+2+2
=9