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+16 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 ______

+33 votes

Best answer

+22 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.

+3 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

0 votes

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.

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