Solved Exercises on Multiplication Factor Worksheets in Grade 3

Master multiplication factors: identifying factors, finding missing factors, factor pairs, and problem-solving strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying factors
Exercise 1
List all factors of 12.
Definition:

Factor: A number that divides evenly into another number. When two factors are multiplied together, they produce the original number.

Method:
  1. Start with 1 and the target number (12)
  2. Find pairs of numbers that multiply to equal 12
  3. Continue until you reach the square root of the number
  4. List all unique factors found
Target
Find factors of 12
Factor Pairs
1×12, 2×6, 3×4
List
1, 2, 3, 4, 6, 12
Step 1: Start with 1 and 12

\(1 \times 12 = 12\), so 1 and 12 are factors

Step 2: Try 2

\(2 \times 6 = 12\), so 2 and 6 are factors

Step 3: Try 3

\(3 \times 4 = 12\), so 3 and 4 are factors

Step 4: Check 4

We already have 4 from step 3, so stop here

Step 5: Organize factors

List from smallest to largest: 1, 2, 3, 4, 6, 12

Factors of 12: 1, 2, 3, 4, 6, 12
Final answer:

The factors of 12 are: 1, 2, 3, 4, 6, 12

Applied rules:

• Factor Pair Rule: If \(a \times b = n\), then both \(a\) and \(b\) are factors of \(n\)

• Divisibility: A factor divides the number evenly with no remainder

• Systematic Approach: Check numbers up to the square root of the target

Factor Tree for 12:

12

/ \

3 4

/ \

2 2

So 12 = 3 × 2 × 2

2 Finding missing factors
Exercise 2
Find the missing factor: \(7 \times ? = 28\)
Definition:

Missing Factor: The unknown number in a multiplication equation that makes the equation true.

Equation
\(7 \times ? = 28\)
Division
\(28 ÷ 7 = ?\)
Result
\(4\)
Step 1: Set up the equation

\(7 \times ? = 28\)

Step 2: Use division to find the missing factor

To find the missing factor, divide the product by the known factor

\(28 ÷ 7 = 4\)

Step 3: Verify the answer

\(7 \times 4 = 28\) ✓

Step 4: Alternative method - repeated subtraction

How many groups of 7 are in 28? Count: 7, 14, 21, 28 → 4 groups

\(7 \times 4 = 28\)
Final answer:

The missing factor is 4

Applied rules:

• Multiplication Division Relationship: \(a \times b = c\) means \(c ÷ a = b\)

• Missing Factor Strategy: Divide the product by the known factor

• Verification: Always check by multiplying back

Visual Model for 7 × ? = 28:

● ● ● ● ● ● ● (Group 1)

● ● ● ● ● ● ● (Group 2)

● ● ● ● ● ● ● (Group 3)

● ● ● ● ● ● ● (Group 4)

Total: 28 dots in 4 groups of 7

3 Factor pairs
Exercise 3
Find all factor pairs of 24.
Definition:

Factor Pair: Two numbers that multiply together to give the target number.

Target
Factor pairs of 24
Pairs
(1,24), (2,12), (3,8), (4,6)
Complete List
All factors: 1,2,3,4,6,8,12,24
Step 1: Start with 1

\(1 \times 24 = 24\), so (1, 24) is a factor pair

Step 2: Try 2

\(2 \times 12 = 24\), so (2, 12) is a factor pair

Step 3: Try 3

\(3 \times 8 = 24\), so (3, 8) is a factor pair

Step 4: Try 4

\(4 \times 6 = 24\), so (4, 6) is a factor pair

Step 5: Check 5

24 ÷ 5 = 4.8, which is not a whole number, so 5 is not a factor

Factor pairs of 24: (1,24), (2,12), (3,8), (4,6)
Final answer:

The factor pairs of 24 are: (1,24), (2,12), (3,8), (4,6)

Applied rules:

• Pair Formation: Each factor has a partner that multiplies to the target

• Efficiency: Only check up to the square root to find all pairs

• Completeness: Once you pass the square root, you've found all pairs

Factor Pairs of 24:

(1, 24): 1 row of 24 OR 24 rows of 1

(2, 12): 2 rows of 12 OR 12 rows of 2

(3, 8): 3 rows of 8 OR 8 rows of 3

(4, 6): 4 rows of 6 OR 6 rows of 4

Factor Concepts & Properties
\(a \times b = c\)
Factor Relationship
Factor Definition
If \(a \times b = c\), then \(a\) and \(b\) are factors of \(c\)
Numbers that divide evenly
Factor Pair
Two numbers that multiply to give the target number
Like (3, 8) for 24
Finding Missing Factor
If \(a \times ? = c\), then \(? = c ÷ a\)
Use division to solve
Every number has at least two factors: 1 and itself
Factor pairs come in pairs: If (a,b) is a pair, so is (b,a)
Tip 1: Use divisibility rules to quickly identify potential factors.
Tip 2: Only check numbers up to the square root of your target number.
Tip 3: To find missing factors, use division: divide the product by the known factor.
Tip 4: Always verify your answer by multiplying the factors back together.
Solution: Exercises 4 to 5
4 Word problem with factors
Exercise 4
Maria has 18 stickers. She wants to arrange them in equal rows with no stickers left over. What are all the possible arrangements she can make?
Definition:

Equal Arrangements: Using factor pairs to determine how objects can be organized in equal rows/columns.

Problem
18 stickers in equal rows
Find
Factors of 18
Arrangements
Based on factor pairs
Step 1: Find all factors of 18

\(1 \times 18 = 18\), so 1 and 18 are factors

\(2 \times 9 = 18\), so 2 and 9 are factors

\(3 \times 6 = 18\), so 3 and 6 are factors

Step 2: List factor pairs

(1, 18), (2, 9), (3, 6)

Step 3: Interpret as arrangements

- 1 row of 18 stickers OR 18 rows of 1 sticker

- 2 rows of 9 stickers OR 9 rows of 2 stickers

- 3 rows of 6 stickers OR 6 rows of 3 stickers

Step 4: Verify each arrangement

\(1 \times 18 = 18\), \(2 \times 9 = 18\), \(3 \times 6 = 18\) ✓

Possible arrangements: (1×18), (2×9), (3×6), (6×3), (9×2), (18×1)
Final answer:

Maria can arrange her stickers in 6 different ways: 1 row of 18, 2 rows of 9, 3 rows of 6, 6 rows of 3, 9 rows of 2, or 18 rows of 1.

Applied rules:

• Real-world Application: Connect factors to practical arrangements

• Systematic Thinking: Use factor pairs to find all possibilities

• Verification: Check that each arrangement uses all 18 stickers

5 Advanced factor identification
Exercise 5
Which of these numbers are factors of 36: 2, 3, 4, 5, 6, 7, 8, 9?
Definition:

Factor Testing: Checking if a number divides another number evenly with no remainder.

Test
Which divide 36 evenly?
Results
2,3,4,6,9 are factors
Not factors
5,7,8 are not
Step 1: Test each number individually

2: \(36 ÷ 2 = 18\) → Yes, 2 is a factor

3: \(36 ÷ 3 = 12\) → Yes, 3 is a factor

4: \(36 ÷ 4 = 9\) → Yes, 4 is a factor

Step 2: Continue testing

5: \(36 ÷ 5 = 7.2\) → No, 5 is not a factor

6: \(36 ÷ 6 = 6\) → Yes, 6 is a factor

7: \(36 ÷ 7 = 5.14...\) → No, 7 is not a factor

Step 3: Final check

8: \(36 ÷ 8 = 4.5\) → No, 8 is not a factor

9: \(36 ÷ 9 = 4\) → Yes, 9 is a factor

Step 4: Verify factor pairs

2×18=36, 3×12=36, 4×9=36, 6×6=36, 9×4=36

Factors of 36 from the list: 2, 3, 4, 6, 9
Final answer:

The factors of 36 from the given list are: 2, 3, 4, 6, and 9

Applied rules:

• Division Test: If division results in a whole number, it's a factor

• Systematic Verification: Test each possibility methodically

• Multiple Approaches: Use both division and multiplication to verify

Multiplication Factors Mastery Guide
\(a \times b = c\)
Factor Relationship
Key definitions:

Factor: A number that divides another number evenly

Factor Pair: Two numbers that multiply to give a specific product

Product: The result of multiplication

Missing Factor: An unknown number in a multiplication equation

Complete methodology:
  1. Identify the Task: Determine if you need to find factors, missing factors, or factor pairs
  2. Apply Appropriate Method: Use division, multiplication, or systematic listing
  3. Solve Systematically: Work in order to avoid missing any possibilities
  4. Verify Solutions: Check answers by multiplying back
  5. Organize Results: Present answers in a clear, logical format
Tip 1: Use divisibility rules to quickly eliminate non-factors (e.g., if a number ends in 5, it's divisible by 5).
Tip 2: For large numbers, only check divisors up to the square root to find all factors efficiently.
Tip 3: Remember that every number is divisible by 1 and itself.
Tip 4: To find missing factors, always use division: divide the product by the known factor.
Common errors: Forgetting to check all numbers up to the square root, missing factor pairs, not verifying answers.
Success strategies: Use systematic approaches, verify all answers, practice with smaller numbers first.
Key rules to remember:

• If \(a \times b = c\), then both \(a\) and \(b\) are factors of \(c\)

• To find missing factors: if \(a \times ? = c\), then \(? = c ÷ a\)

• Every number has at least two factors: 1 and itself

• Prime numbers have exactly two factors: 1 and themselves

• Composite numbers have more than two factors

Factor Chart Reference
Essential Factor Charts
Common factor charts for multiplication practice:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Patterns to notice:

  • All numbers have 1 and themselves as factors
  • Even numbers always have 2 as a factor
  • Numbers ending in 5 or 0 have 5 as a factor
  • Square numbers have an odd number of factors

Questions & Answers

Question: How do I know if a number is a factor of another number?

Answer: A number is a factor of another number if it divides evenly with no remainder. Here are two ways to check:

  • Division Method: Divide the larger number by the smaller number. If there's no remainder, it's a factor.
  • Multiplication Method: See if the smaller number times another whole number equals the larger number.

For example, to check if 4 is a factor of 12:

  • Division: \(12 ÷ 4 = 3\) (no remainder) → Yes, 4 is a factor
  • Multiplication: \(4 \times 3 = 12\) → Yes, 4 is a factor

If there's a remainder, it's not a factor. For example: \(13 ÷ 4 = 3\) remainder 1 → 4 is not a factor of 13.

Question: My child is having trouble finding all the factors of a number. What's the best approach?

Answer: Use a systematic approach to find all factors:

  1. Start with 1: 1 is always a factor
  2. Work systematically: Check 2, 3, 4, etc. in order
  3. Find factor pairs: If 2 goes into the number, write down both 2 and its pair
  4. Stop at the square root: Once you pass the square root, you'll just repeat factors
  5. Check divisibility: Use divisibility rules to speed up the process

For example, finding factors of 24:

  • 1 and 24 (since \(1 \times 24 = 24\))
  • 2 and 12 (since \(2 \times 12 = 24\))
  • 3 and 8 (since \(3 \times 8 = 24\))
  • 4 and 6 (since \(4 \times 6 = 24\))

After 4 (which is less than √24 ≈ 4.9), we'd get 6 again, so we stop. The factors are: 1, 2, 3, 4, 6, 8, 12, 24.

Question: What's the difference between a factor and a multiple?

Answer: Factors and multiples are related but opposite concepts:

  • Factor: A number that divides into another number evenly. It's usually smaller than the original number.
  • Multiple: The result of multiplying a number by a whole number. It's usually larger than the original number.

For example, using the number 6:

  • Factors of 6: 1, 2, 3, 6 (these divide into 6 evenly)
  • Multiples of 6: 6, 12, 18, 24, 30, ... (these are 6 times 1, 2, 3, 4, 5, ...)

Think of it this way: factors go INTO the number, while multiples come FROM the number.

If 2 × 3 = 6, then 2 and 3 are factors of 6, and 6 is a multiple of both 2 and 3.