GATE CS 1995 Q2.12 — Number Representations (Fixed and Floating Point)

We can find the number of $1$s efficiently by converting the given expression into its hexadecimal (base-16) representation.

Expressing the given terms as powers of $16$:

• $4096 = 16^3$
• $256 = 16^2$
• $16 = 16^1$
• $1 = 16^0$

Substituting these into the expression:

In hexadecimal, this is represented as $(3\text{F}53)_{16}$ (since $15$ corresponds to $\text{F}$ in hex).

Now, convert each hexadecimal digit to its 4-bit binary equivalent and count the number of $1$s:

• $3_{16} = 0011_2$ → contains 2 ones
• $\text{F}_{16} = 1111_2$ → contains 4 ones
• $5_{16} = 0101_2$ → contains 2 ones
• $3_{16} = 0011_2$ → contains 2 ones

Total number of $1$s = $2 + 4 + 2 + 2 = 10$.