# GATE2010-8

5.9k views

$P$ is a $16$-bit signed integer. The $2$'s complement representation of $P$ is $(F87B)_{16}$. The $2$'s complement representation of $8\times P$ is

1. $(C3D8)_{16}$
2. $(187B)_{16}$
3. $(F878)_{16}$
4. $(987B)_{16}$

edited
9

$Remark:$

One arithmetic shift to left multiplies the number by 2, but we lose left most bits one by one

(AshiftL#2, R0)

PS: Arithmetic shift to left is same as Logical shift to left.

1

Left shift moves the MSB to $2x$ the weight. Doing it thrice would give us $8x$ the weight.

Multiplication can be directly carried in 2's complement form. F87B = 1111 1000 0111 1011 can be left shifted 3 times to give 8P = 1100 0011 1101 1000 = C3D8.

Or, we can do as follows:

MSB in (F87B) is 1. So, P is a negative number. So, P = -1 * 2's complement of (F87B) = -1 * (0785) = -1 * (0000 0111 1000 0101)

8 * P = -1 *  (0011 1100 0010 1000) (P in binary left shifted 3 times)

In 2's complement representation , this equals, 1100 0011 1101 1000 = C3D8

Correct Answer: $A$

edited
8
The first one was quite easier. :)
0
why binary left shift why not right shift ?
17
Left shift means adding a 0 at right end, which is equivalent to multiply by 2 in binary. Right shift will be doing division by 2.
1
ya rgt thanks:)
1
suppose iif the question asks 10*P then ??
22
@Puja you can do it  (10*P = 8*P + 2*P) where (8*P) you shift 3 digit left and then for (2*P) you can shift it 1 digit left and then add it.
2
sir I want to know why should be shift 3 times ?

if it is 9 then ?
1
@hem chandra

Left shift  is equal to multiplication and right shift equals to division in binary numbers, for the factor of 2 like 2 , 2^2, 2^3 .............

Since here it is 8 , it is 2^3. So we need to shift 3 times
4
if there is 9p then 8p+1p = 2^3p+2^0p (overall three left shifts)
1
Thank You so much for this explanation :)
1

If you do 10*P then You will get

= 8*P + 2*P

= 1 1011 0100 1100 1110

This is the Case of Overflow as said in Question P is 16 Bit Signed Integer.

So i guess there will be one option of overflow in this scenario.

0
@krishn.jh It won't be overflow because for overflow $C_{in} \neq C_{out}$ but here $C_{in} = C_{out} = 1$
0
@Akhilesh so you mean to say

if you do 8*P + 2*P then no overflow will occur?
0
It will be end carry, which is discarded in 2's complement.

And in 2's complement we can forget all 1s in MSBs, except one 1.
Eg. 111011 = 1011

Thus, we can forget the end carry in our result of 8P+2P because sign(negative here) as well as value is still preserved.
0
@Akhilesh ok yes i know as concept says preceding 1's does not affect the values in 2's Complement as they make one partial block.

i got it as after discarding the carry 1, it is not going to affect the final result.

But Be careful  as if sum is 0 carry 1 in MSB place then surely there would be Overflow.
0
1010=10

then after left shift by 1 bit it becomes 0100=4 ,why didnt i get 20?
0
0
Can we do division directly on 2's complement form also?
0
shifting mechanism can't be applied in case of 13*P, right ??
0

@hem  Does it mean that the value of 8p and 9p will be same ? Please clear ,thank you !

$P=(F87B)_{16}=(1111|1000|0111|1011)_{2}:$ 2's compliment representation

What is this number ?

$(0000|0111|1000|0100)_{2}+1=(0000|0111|1000|0101)_{2}=1925$

$So, P\ is =-1925$

$-1925\times 8=-15400$

Find 2's compliment representation of $-15400$

$+15400=(0011|1100|0010|1000)_{2}$

$(1100|0011|1101|0111)_{2}+1=(1100|0011|1101|1000)_{2}=(C3D8)_{16}$

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