Shifts as multiplying and dividing by powers of 2

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Quick Questions

Shade in the A or B circle for each question.

1. A logical left shift of 3 places multiplies a binary number by what?

A 8

B 6

2. What is 00000101 (5) after a logical left shift of 3 places?

A 00101000 (40)

B 00001111 (15)

3. What is 00011100 (28) after a logical right shift of 2 places?

A 00001110 (14)

B 00000111 (7)

4. A right shift of 2 places is applied to 00001111 (15). What is the result?

A 00000011 (3)

B 3.75

5. Which statement about a right shift by 2 places is correct?

A It divides by 2 and keeps the remainder

B It divides by 4 and drops any remainder

Match the Pairs

Draw a line connecting each item on the left to its match on the right.

Left column		Right column
100000101 << 3		ABinary shifts cannot make fractions. The remainder is dropped, so 13 / 4 gives 3 (00000011), not 3.25.
200010100 >> 2		BShifts right 2 places, dividing 20 by 2^2 (4) to give 5 (00000101)
3A right shift by 2 divides by 4 and drops the remainder, so 15 becomes 3		C11000000 << 1 = 10000000
4A left shift loses the answer if a 1 is shifted off the end of the 8-bit register		DOnly shifted left 2 places, which multiplies by 4. Shifting left 3 places multiplies by 8, so the answer is 24 (00011000).
500000011 << 3 = 00001100 (3 x 8 = 12)		EShifts left 3 places, multiplying 5 by 2^3 (8) to give 40 (00101000)
600001101 >> 2 = 00000011.25 (13 / 4)		F00001111 >> 2 = 00000011

Definitions

n shifts = 2^n

Fill in the Blanks

Complete each line of code by filling in the missing parts.

Left shift by 3 multiplies by _

Right shift by 2 divides by _

00000110 << 3 = ________

13 >> 2 is 13 / 4 = 3 remainder _

10000001 << 1 = 00000010, so the 1 is ____

Spot the Error

Shade in the letter of the line that contains an error.

Question 1

A Start: 00000101 (5)

B Shift left 3 places

C Multiply by 2^3 = 8

D Result: 5 x 8 = 48

Question 2

A Start: 00001100 (12)

B Shift left 2 places

C Divide by 2^2 = 4

D Result: 00000011 (3)

Question 3

A x = 00000011 (5)

B Shift left 3 places

C Multiply by 2^3 = 8

D Result: 00011000 (24)

Question 4

A Right shift 2: multiply by 4

B Start: 00010100 (20)

C Result: 00000101 (5)

Question 5

A Start: 00001101 (13)

B Shift right 2 places

C 13 / 4 = 3 remainder 1

D Result: 00000100 (4)

Answer Key

Shifts as multiplying and dividing by powers of 2

Section A: Quick Questions

A — 8

A — 00101000 (40)

B — 00000111 (7)

A — 00000011 (3)

B — It divides by 4 and drops any remainder

Section B: Match the Pairs

00000101 << 3 → Shifts left 3 places, multiplying 5 by 2^3 (8) to give 40 (00101000)

00010100 >> 2 → Shifts right 2 places, dividing 20 by 2^2 (4) to give 5 (00000101)

A right shift by 2 divides by 4 and drops the remainder, so 15 becomes 3 → 00001111 >> 2 = 00000011

A left shift loses the answer if a 1 is shifted off the end of the 8-bit register → 11000000 << 1 = 10000000

00000011 << 3 = 00001100 (3 x 8 = 12) → Only shifted left 2 places, which multiplies by 4. Shifting left 3 places multiplies by 8, so the answer is 24 (00011000).

00001101 >> 2 = 00000011.25 (13 / 4) → Binary shifts cannot make fractions. The remainder is dropped, so 13 / 4 gives 3 (00000011), not 3.25.

Section C: Definitions

n shifts = 2^n — Each left shift multiplies a binary number by 2 and each right shift divides it by 2, so n shifts multiply or divide by 2^n. Bits shifted off the end are lost.

Section D: Fill in the Blanks

Left shift by 3 multiplies by 8

Right shift by 2 divides by 4

00000110 << 3 = 00110000

13 >> 2 is 13 / 4 = 3 remainder 1

10000001 << 1 = 00000010, so the 1 is lost

Section E: Spot the Error

D — 5 x 8 = 40, not 48. Three left shifts multiply by 2^3 = 8, so 5 becomes 40 (00101000).

B — The result 3 comes from dividing by 4, which needs a RIGHT shift of 2 places. A left shift would multiply by 4. It should read 'Shift right 2 places'.

A — 00000011 is 2 + 1 = 3, not 5. The start value must be 3 so that 3 x 8 = 24 matches the result 00011000.

A — A right shift divides the number. Two right shifts divide by 2^2 = 4, so 20 becomes 5. It should say 'divide by 4'.

D — A right shift drops the remainder, it does not round up. 13 / 4 = 3 remainder 1, so the result is 00000011 (3).