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The number of $1$'s in the binary representation of $(3*4096 + 15*256 + 5*16 + 3)$ are:

1. $8$
2. $9$
3. $10$
4. $12$

edited | 7.2k views

I suggest following approach , here we can clearly see that numbers are getting multiplied by powers of 16. So this is nothing but Hexadecimal number in disguise.

$(3\times 4096 + 15 \times 256 + 5 \times 16 + 3) = (3F53)_{16} = (0011111101010011)_2$ which has total $2 + 4 + 2 + 2 = 10$ 1's

Correct Answer: $C$
by Boss (41.9k points)
edited
+4
awesome

We have,  3*4096 + 15*256 + 5*16 + 3

= (2+1)*212     + (8+4+2+1) * 28          +  (4+1)*24  + 2 + 1

= 213 + 212      + 211  + 210 + 2+ 28   +   26 + 24   + 2 + 1

= 1        1            1       1        1      1          1      1      1     1

= 10  1's     So, OPTION (C) ..

by Boss (15.9k points)
0
Thanks @Himanshu it is best approach.
$3 = (11)_2$
$3\times 4096 = 3\times (2^{12}) = (11)_2<< 12 = (11000000000000)_2$

Similarly, $15 \times 256 = (1111)_2 << 8 = (111100000000)_2$ and $5 \times 16 = (101)_2 << 4 = (1010000)_2$

So, $3\times 4096 + 15 \times 256 + 5\times 16 + 3 = (11111101010011)_2$

Number of 1's = 10.
by Veteran (431k points)
+1
best approach..

All 4096,256,16 needs only 1 one to be represented in binary

3 - requires 2   1's

15 - requires 4 1's

5 - requires 2 1's

3 - requires 2 1's

so adding all those we get 2+4+2+2= 10
by Active (3.5k points)
+3
10 is correct but the approach works only if the numbers doesn't have a 1 in the same place. You have to take care of that.

((3*4096) +(15*256) + (5*16) + 3 ) = 16211

= (11111101010011)2

by (319 points)

Powers of 16 go like: 1,16, 256, 1024, 65536, 1048576...

Interestingly enough, these are all powers of 2 as well.

As it happens, even the powers of 8 conform to this pattern.

Powers of 8 go like: 1, 8, 64, 512, 4096, 32768

That's because 16 is nothing but $2^4$; so $16^2$ = $(2^4)^2$ = $2^8$. Same can be derived for 8.

So, when you see 4096, don't immediately jump to $2^{12}$ — first check if it is a power of 16 or 8 as well. Because higher the base, easier is to solve such questions.

Coming to the question, given is hexadecimal format => 0x3F53 = Ten 1's in binary.

Option C

by Loyal (6.6k points)