Question

let m be a positive integer with n bit binary representation an-1 an-2 ... a1a0 with an-1=1 what are the smallest and largest values that m could have

Answer 1

Answer:

Explanation:

From the given information:

$a_{n-1} , a_{n-2}...a_o$ in binary is:

$a_{n-1}\times 2^{n-1} + a_{n-2}}\times 2^{n-2}+ ...+a_o$

So, the largest number posses all $a_{n-1} , a_{n-2}...a_o$  nonzero, however, the smallest number has $a_{n-2} , a_{n-3}...a_o$ all zero.

∴

The largest = 11111. . .1 in n times and the smallest = 1000. . .0 in n -1 times

i.e.

$(11111111...1)_2 = ( 1 \times 2^{n-1} + 1\times 2^{n-2} + ... + 1 )_{10}$

$= \dfrac{1(2^n-1)}{2-1}$

$\mathbf{=2^n -1}$

$(1000...0)_2 = (1 \times 2^{n-1} + 0 \times 2^{n-2} + 0 \times 2^{n-3} + ... + 0)_{10}$

$\mathbf {= 2 ^{n-1}}$

Hence, the smallest value is $\mathbf{2^{n-1}}$ and the largest value is $\mathbf{2^{n}-1}$