Solved Exercises on Finding Factor Pairs in Grade 4

Master finding factor pairs: identifying factor pairs systematically, organizing them, and applying them to solve mathematical problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding factor pairs
Exercise 1
Find all factor pairs of 12. List them in order from smallest to largest first factor.
Definition:

Factor pair: Two numbers that multiply together to give the original number

Example: For 12, a factor pair is 3 and 4 because 3 × 4 = 12

Factor pair method:
  1. Start with 1 and the number itself
  2. Check each number up to the square root of the number
  3. For each divisor found, record both the divisor and its pair
  4. Stop when factors start repeating
  5. Write pairs in order
Number to factor
12
Factor pairs
(1,12), (2,6), (3,4)
Step 1: Start with 1

12 ÷ 1 = 12, so (1, 12) is a factor pair

Step 2: Check 2

12 ÷ 2 = 6, so (2, 6) is a factor pair

Step 3: Check 3

12 ÷ 3 = 4, so (3, 4) is a factor pair

Step 4: Check 4

12 ÷ 4 = 3, but we already have (3, 4), so we stop here

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

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

Applied rules:

• Factor pair definition: Two numbers that multiply to the original number

• Systematic approach: Check divisors in order to avoid missing any

• Stopping condition: Stop when factors repeat to avoid duplicates

2 Organizing factor pairs
Exercise 2
Find and organize the factor pairs of 18 in a table format.
Definition:

Organized factor pairs: Listing factor pairs in a systematic way, often in a table

Number to factor
18
Factor pairs
(1,18), (2,9), (3,6)
Step 1: Check 1

18 ÷ 1 = 18, so (1, 18) is a factor pair

Step 2: Check 2

18 ÷ 2 = 9, so (2, 9) is a factor pair

Step 3: Check 3

18 ÷ 3 = 6, so (3, 6) is a factor pair

Step 4: Check 4

18 ÷ 4 = 4.5 (not a whole number), so 4 is not a factor

Step 5: Check 5

18 ÷ 5 = 3.6 (not a whole number), so 5 is not a factor

Step 6: Check 6

18 ÷ 6 = 3, but we already have (3, 6), so we stop

First Factor Second Factor Product
1 18 1 × 18 = 18
2 9 2 × 9 = 18
3 6 3 × 6 = 18
Factor pairs of 18: (1, 18), (2, 9), (3, 6)
Final answer:

The factor pairs of 18 are: (1, 18), (2, 9), (3, 6)

Applied rules:

• Organization: Tables help visualize relationships between factors

• Verification: Check that each pair multiplies to the original number

• Completeness: Ensure all factor pairs are included

3 Prime factorization
Exercise 3
Find the prime factorization of 24 using factor pairs.
Definition:

Prime factorization: Expressing a number as a product of prime numbers

Prime number: A number with exactly two factors: 1 and itself

Start with
24
Break down
2 × 12
Continue
2 × 2 × 6
Final
2³ × 3
Step 1: Find first factor pair

Start with 24 = 2 × 12 (since 2 is the smallest prime factor)

Step 2: Factor 12

12 = 2 × 6

Step 3: Factor 6

6 = 2 × 3

Step 4: Check if all factors are prime

2 is prime, 3 is prime

Step 5: Write prime factorization

24 = 2 × 2 × 2 × 3 = 2³ × 3

24
2 × 12
2 × 2 × 6
2 × 2 × 2 × 3
Prime factorization of 24: 2³ × 3
Final answer:

The prime factorization of 24 is 2³ × 3.

Applied rules:

• Prime factorization: Keep factoring until all factors are prime

• Prime numbers: Numbers with exactly two factors (1 and itself)

• Exponential form: Write repeated prime factors as powers

Factor Pairs Fundamentals
\(\text{Factor}_1 \times \text{Factor}_2 = \text{Number}\)
Factor Pair Equation
Key definitions:

Factor pair: Two numbers that multiply to give a specific number

Factor: A number that divides another number evenly with no remainder

Prime number: A number with exactly two factors: 1 and itself

Composite number: A number with more than two factors

Prime factorization: Writing a number as a product of prime numbers

Perfect square: A number that has a factor pair where both factors are the same

Greatest Common Factor (GCF): The largest factor shared by two or more numbers

Systematic factor pair finding:
  1. Start with 1: Every number has (1, number) as a factor pair
  2. Check sequentially: Test 2, 3, 4, etc. up to √n
  3. Record pairs: When you find a factor, record both numbers
  4. Stop efficiently: Stop when factors repeat
  5. Organize: List pairs in order from smallest first factor
Tip 1: Check only up to the square root to avoid duplicates.
Tip 2: If you find factor a, then n/a is also a factor.
Tip 3: Perfect squares have one factor pair where both numbers are the same.
Tip 4: Use divisibility rules to quickly identify potential factors.
Common errors: Missing factor pairs, including non-factors, not stopping at the square root.
Key insight: Factor pairs always come in pairs unless it's a perfect square.
Divisibility rules:

• Divisible by 2: Number ends in 0, 2, 4, 6, or 8

• Divisible by 3: Sum of digits is divisible by 3

• Divisible by 4: Last two digits form a number divisible by 4

• Divisible by 5: Number ends in 0 or 5

• Divisible by 6: Number is divisible by both 2 and 3

• Divisible by 9: Sum of digits is divisible by 9

• Divisible by 10: Number ends in 0

Perfect Squares and Their Factor Pairs

Perfect Squares

Perfect squares have an odd number of factors because one factor pair has identical numbers:

4: (1,4), (2,2) → 3 factors

9: (1,9), (3,3) → 3 factors

16: (1,16), (2,8), (4,4) → 5 factors

Regular Numbers

Non-perfect squares have an even number of factors:

6: (1,6), (2,3) → 4 factors

8: (1,8), (2,4) → 4 factors

10: (1,10), (2,5) → 4 factors

Solution: Exercises 4 to 5
4 Application problem
Exercise 4
Maria wants to arrange 20 chairs in rows with the same number of chairs in each row. What are all the possible arrangements she can make?
Definition:

Arrangement problem: Using factor pairs to determine possible configurations

Number of chairs
20
Factor pairs
(1,20), (2,10), (4,5)
Possible arrangements
See solution
Step 1: Find factor pairs of 20

20 ÷ 1 = 20, so (1, 20)

20 ÷ 2 = 10, so (2, 10)

20 ÷ 4 = 5, so (4, 5)

20 ÷ 5 = 4, but we already have (4, 5)

Step 2: Interpret factor pairs as arrangements

(1, 20): 1 row of 20 chairs OR 20 rows of 1 chair

(2, 10): 2 rows of 10 chairs OR 10 rows of 2 chairs

(4, 5): 4 rows of 5 chairs OR 5 rows of 4 chairs

Step 3: List all possible arrangements

1 row of 20 chairs

20 rows of 1 chair

2 rows of 10 chairs

10 rows of 2 chairs

4 rows of 5 chairs

5 rows of 4 chairs

1 × 20

20 rows of 1

2 × 10

10 rows of 2

4 × 5

5 rows of 4

6 possible arrangements:
1×20, 2×10, 4×5, 5×4, 10×2, 20×1
Final answer:

Maria can arrange the chairs in 6 ways:

• 1 row of 20 chairs

• 20 rows of 1 chair

• 2 rows of 10 chairs

• 10 rows of 2 chairs

• 4 rows of 5 chairs

• 5 rows of 4 chairs

Applied rules:

• Application: Factor pairs help solve real-world arrangement problems

• Interpretation: Each factor pair can represent different configurations

• Completeness: Consider both orientations of each factor pair

5 Advanced factor pairs
Exercise 5
Find all factor pairs of 36 and identify which pair has the smallest difference between the two factors.
Definition:

Factor pair with smallest difference: The pair where the two numbers are closest to each other

Number to factor
36
All factor pairs
(1,36), (2,18), (3,12), (4,9), (6,6)
Smallest difference
(6,6)
Step 1: Find all factor pairs of 36

36 ÷ 1 = 36, so (1, 36)

36 ÷ 2 = 18, so (2, 18)

36 ÷ 3 = 12, so (3, 12)

36 ÷ 4 = 9, so (4, 9)

36 ÷ 6 = 6, so (6, 6)

Step 2: Calculate differences between factors

(1, 36): 36 - 1 = 35

(2, 18): 18 - 2 = 16

(3, 12): 12 - 3 = 9

(4, 9): 9 - 4 = 5

(6, 6): 6 - 6 = 0

Step 3: Identify the pair with smallest difference

(6, 6) has the smallest difference of 0

(1, 36)
(2, 18)
(3, 12)
(4, 9)
(6, 6)
Factor pairs of 36: (1,36), (2,18), (3,12), (4,9), (6,6)
Smallest difference: (6,6)
Final answer:

All factor pairs of 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)

The pair with the smallest difference is (6, 6).

Applied rules:

• Factor pairs: List all pairs systematically

• Difference calculation: Subtract the smaller factor from the larger

• Perfect square: 36 is a perfect square, so it has a factor pair with identical numbers

Advanced Factor Pair Concepts
\(\text{If } a \times b = n \text{, then } a \text{ and } b \text{ are factor pairs of } n\)
Factor Pair Definition
Key definitions:

Factor pair: Two numbers that multiply to give a specific number

Prime factorization: Breaking down a number into its prime factors

Greatest Common Factor (GCF): Largest factor shared by two or more numbers

Least Common Multiple (LCM): Smallest multiple shared by two or more numbers

Perfect square: Number with a factor pair where both factors are equal

Relatively prime: Two numbers whose GCF is 1

Abundant number: Number whose sum of proper factors exceeds the number itself

Advanced factor pair applications:
  1. Rectangle dimensions: Factor pairs represent possible length-width combinations
  2. Grouping problems: Factor pairs determine possible group sizes
  3. Optimization: Factor pairs help find arrangements with minimal difference
  4. Number classification: Factor pairs help identify perfect squares and primes
Tip 1: The factor pair with the smallest difference is closest to √n.
Tip 2: Perfect squares always have an odd number of total factors.
Tip 3: Prime numbers have only one factor pair: (1, the number).
Tip 4: For large numbers, use divisibility rules to find factors efficiently.
Special cases: 1 has only one factor (itself), primes have exactly 2 factors.
Efficiency: Check only up to √n to find all factor pairs.
Advanced relationships:

• Factor count formula: If prime factorization is p^a × q^b, then number of factors is (a+1)(b+1)

• GCF-LCM relationship: GCF(a,b) × LCM(a,b) = a × b

• Factor symmetry: Factor pairs come in symmetric pairs around √n

• Perfect square test: A number is a perfect square if it has an odd number of factors

Prime Factorization Trees

Example: Factor 24
24
2 × 12
2 × 2 × 6
2 × 2 × 2 × 3

Prime factorization: 24 = 2³ × 3

Prime factors: 2, 2, 2, 3

Example: Factor 30
30
2 × 15
2 × 3 × 5

Prime factorization: 30 = 2 × 3 × 5

Prime factors: 2, 3, 5

Questions & Answers

Question: I keep forgetting which numbers to check when finding factor pairs. Is there a shortcut?

Answer: Yes, there are some great shortcuts!

  • Check only up to the square root: If you're finding factors of 36, you only need to check up to 6 (√36 = 6)
  • Use divisibility rules: Check if the number is even (divisible by 2), if digits sum to a multiple of 3 (divisible by 3), etc.
  • Start with small primes: Check 2, 3, 5, 7 first since they're the most common factors

Remember: once you pass the square root, you'll just find the reverse of factor pairs you've already found!

For example, for 24: once you check up to 4 (since √24 ≈ 4.9), you'll have found (1,24), (2,12), (3,8), (4,6). Checking 6 would just give you (6,4), which is the same pair!

Question: My child is getting confused between factors and multiples. How can I help them understand the difference?

Answer: This is a very common confusion! Here's how to clarify:

  • Factors: "Go into" the number (divide it). Factors are usually smaller than or equal to the number. Example: factors of 12 are 1, 2, 3, 4, 6, 12.
  • Multiples: "Come from" the number (are made by multiplying it). Multiples are usually larger than or equal to the number. Example: multiples of 3 are 3, 6, 9, 12, 15, 18...

A helpful memory trick: FFactors go into the number (they divide it), and MMultiples come from the number (they are made by multiplying it).

Also remember: There are a limited number of factors but infinitely many multiples!

Use visual aids: factors are like "parts" of the number, multiples are like "extensions" of the number.

Question: How do factor pairs relate to other mathematical concepts students will encounter later?

Answer: Factor pairs are foundational to many advanced concepts:

  • Greatest Common Factor (GCF): Students need to identify common factors to find the GCF
  • Least Common Multiple (LCM): Understanding multiples helps find the LCM
  • Prime factorization: Breaking numbers into factor pairs leads to prime factorization
  • Fraction simplification: Finding common factors helps reduce fractions
  • Algebra: Factoring quadratic expressions uses similar principles
  • Geometry: Finding dimensions of rectangles with given area

Factor pairs also introduce the fundamental theorem of arithmetic (every integer has a unique prime factorization) and prepare students for number theory concepts.

Additionally, understanding factor pairs helps with divisibility rules, which are essential for working with fractions and ratios.