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+11 votes

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 ______

$$16^3*9 + 16^2*7 + 16*5+3$$

The number of $1’s$ in the unsigned binary representation of the number is ______

+24 votes

Best answer

+4 votes

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.

0 votes

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

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

0 votes

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

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

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