ITPEC FE Subject A October 2025 Question 2
Let n be a binary integer represented in two’s complement. Which of the following is the operation that results in the value 9 × n using only bit shifting and an addition or subtraction?
Explanation
Shift 3 bits to the left, then add the original value — 9n = 8n + n, and a 3-bit left shift multiplies by 8.
A left shift by k bits multiplies a two's complement integer by 2k, as long as the result does not overflow.
- •Write 9 as a power of two plus a small term: 9 = 8 + 1 = 23 + 1
- •Shift n 3 bits to the left: 8n
- •Add n: 8n + n = 9n
Answer (c)
Why not others:
- •(a) — a 2-bit shift is only ×4, so the result is 4n + n = 5n
- •(b) — 4n − n = 3n
- •(d) — subtracts instead of adding: 8n − n = 7n
Key rule: To multiply by a constant with shifts, write it as a sum or difference of powers of two, such as 9 = 23 + 1 or 7 = 23 − 1, and shift once for each power.
Basic Theory · AI-written, may contain errors
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