Applied Problem-Solving in Factors, Multiples, and Patterns

Master applied problem-solving in factors, multiples, and patterns through these 5 detailed exercises with visual infographics and step-by-step solutions.

Solution: Exercises 1 to 3
1 Finding factors
Exercise 1
Sarah has 24 flowers that she wants to arrange in equal rows. In how many different ways can she arrange them? List all possible arrangements.
Definition:

Factors: Numbers that divide evenly into another number. For example, the factors of 24 are numbers that divide 24 without leaving a remainder.

Method to find factors:
  1. Start with 1 and the number itself
  2. Check each number up to the square root of the number
  3. If a number divides evenly, add both the divisor and the quotient
  4. List all unique factors
Problem
Arrange 24 flowers in equal rows
Find factors of 24
1, 2, 3, 4, 6, 8, 12, 24
Step 1: Find all factors of 24

1×24=24, 2×12=24, 3×8=24, 4×6=24

So factors are: 1, 2, 3, 4, 6, 8, 12, 24

Step 2: Interpret factors as possible arrangements

Each factor represents a possible number of rows or flowers per row

1 row of 24 flowers, 2 rows of 12 flowers, 3 rows of 8 flowers, etc.

Step 3: List all possible arrangements

8 different arrangements: 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1

8 different arrangements possible
Final answer:

Sarah can arrange her 24 flowers in 8 different ways:

1 row of 24, 2 rows of 12, 3 rows of 8, 4 rows of 6, 6 rows of 4, 8 rows of 3, 12 rows of 2, or 24 rows of 1

Applied rules:

• Factor identification: Find all numbers that divide 24 evenly

• Application: Each factor pair represents a possible arrangement

• Verification: Multiply each factor pair to ensure it equals 24

🌸
Factor Tree
24
2×12
2×2×6
2×2×2×3
📊
Possible Arrangements
1×24
2×12
3×8
4×6
6×4
2 Finding multiples
Exercise 2
A bus comes every 15 minutes and a train comes every 20 minutes. If both arrive at the station at 8:00 AM, when will they next arrive together?
Definition:

Least Common Multiple (LCM): The smallest positive number that is a multiple of two or more numbers.

Problem
Bus every 15 min, Train every 20 min
Find LCM of 15 and 20
LCM(15, 20) = 60
Next arrival
8:00 AM + 60 min = 9:00 AM
Step 1: List multiples of each number

Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120...

Multiples of 20: 20, 40, 60, 80, 100, 120, 140...

Step 2: Identify the least common multiple

The smallest number that appears in both lists is 60

So LCM(15, 20) = 60

Step 3: Calculate the next simultaneous arrival

They will both arrive together again in 60 minutes

8:00 AM + 60 minutes = 9:00 AM

Both will arrive together at 9:00 AM
Final answer:

The bus and train will next arrive together at 9:00 AM.

Applied rules:

• LCM concept: Find the smallest common multiple of given numbers

• Time calculation: Add LCM minutes to the starting time

• Verification: Check that both intervals fit evenly into the LCM

🚌
Bus Schedule
8:00
8:15
8:30
8:45
9:00
🚂
Train Schedule
8:00
8:20
8:40
9:00
9:20
3 Pattern recognition
Exercise 3
A pattern of tiles follows this sequence: 3, 6, 9, 12, 15, ... If the pattern continues, what will be the 20th number in the sequence? How many tiles are needed for the first 10 figures?
Definition:

Arithmetic sequence: A sequence where each term increases by a constant amount called the common difference.

Pattern
3, 6, 9, 12, 15, ...
Rule
Multiply position by 3
20th term
20 × 3 = 60
Step 1: Identify the pattern rule

3 = 1×3, 6 = 2×3, 9 = 3×3, 12 = 4×3, 15 = 5×3

The nth term is 3n

Step 2: Find the 20th term

Using the rule: 20th term = 3×20 = 60

Step 3: Find sum of first 10 terms

First 10 terms: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

Sum = 3(1+2+3+4+5+6+7+8+9+10) = 3×55 = 165

20th term: 60, Sum of first 10: 165
Final answer:

The 20th number in the sequence is 60.

The first 10 figures require 165 tiles total.

Applied rules:

• Pattern recognition: Identify the rule that generates the sequence

• Formula application: Use the rule to find any term in the sequence

• Series summation: Add consecutive terms using arithmetic series formula

🔢
Sequence Pattern
3
6
9
12
15

Rule: n × 3

🎯
Term Calculation
nth term
n × 3
20 × 3 = 60
Key Concepts and Methods
Factor × Factor = Product
Factors Relationship
Factor Identification
1, 2, 3, 4, 6, 8, 12, 24
Factors of 24
Multiples
15, 30, 45, 60...
Multiples of 15
Pattern Rule
3n
nth term
Key definitions:

Factors: Numbers that divide evenly into another number

Multiples: Products of a number and any integer

Patterns: Sequences that follow a predictable rule

Problem-solving methodology:
  1. Read carefully: Understand what the problem is asking
  2. Identify key words: Look for terms like "arrange," "together," "sequence," "pattern"
  3. Choose method: Determine if it's a factor, multiple, or pattern problem
  4. Solve systematically: Apply the appropriate mathematical operation
  5. Check answer: Verify your solution makes sense
Tip 1: When finding factors, work systematically from 1 up to the square root.
Tip 2: For LCM problems, list multiples until you find a common one.
Tip 3: For patterns, look for the relationship between position and value.
Tip 4: Always verify your answer by substituting back into the original problem.
Solution: Exercises 4 to 5
4 Greatest Common Factor
Exercise 4
Emma has 36 red balloons and 48 blue balloons. She wants to make identical balloon bouquets with the same number of red and blue balloons in each bouquet, using all balloons. What is the greatest number of bouquets she can make? How many balloons of each color will be in each bouquet?
Definition:

Greatest Common Factor (GCF): The largest number that divides two or more numbers without leaving a remainder.

Given
36 red, 48 blue balloons
Find GCF of 36, 48
GCF(36, 48) = 12
Bouquets
12 bouquets
Step 1: Find factors of 36 and 48

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Step 2: Identify common factors

Common factors: 1, 2, 3, 4, 6, 12

Greatest common factor: 12

Step 3: Calculate balloons per bouquet

Red per bouquet: 36 ÷ 12 = 3

Blue per bouquet: 48 ÷ 12 = 4

12 bouquets with 3 red and 4 blue balloons each
Final answer:

Emma can make 12 identical bouquets.

Each bouquet will contain 3 red balloons and 4 blue balloons.

Applied rules:

• GCF application: Use GCF to find maximum number of identical groups

• Division: Divide total items by group count to get items per group

• Verification: Multiply back to ensure original quantities are preserved

🎈
Balloon Distribution
36 red
÷12 bouquets
3 red per bouquet
🔵
Blue Balloons
48 blue
÷12 bouquets
4 blue per bouquet
5 Pattern continuation
Exercise 5
A tile pattern starts with 4 tiles in the first figure, 7 tiles in the second figure, 10 tiles in the third figure. If this pattern continues, how many tiles will be in the 15th figure? Which figure will contain exactly 61 tiles?
Definition:

Linear pattern: A pattern where the number of elements increases by a constant amount each time.

Pattern
4, 7, 10, ...
Rule
3n + 1
15th figure
3(15) + 1 = 46
Step 1: Identify the pattern rule

Figure 1: 4 tiles, Figure 2: 7 tiles, Figure 3: 10 tiles

Each figure adds 3 tiles: 4, 4+3, 4+3+3, ...

Pattern rule: 3n + 1 (where n is the figure number)

Step 2: Find tiles in 15th figure

Using the rule: 3(15) + 1 = 45 + 1 = 46 tiles

Step 3: Find which figure has 61 tiles

Set up equation: 3n + 1 = 61

3n = 60, so n = 20

The 20th figure will have 61 tiles

Step 4: Verify the results

15th figure: 3(15) + 1 = 46 ✓

20th figure: 3(20) + 1 = 61 ✓

15th figure: 46 tiles, 20th figure: 61 tiles
Final answer:

The 15th figure will contain 46 tiles.

The 20th figure will contain exactly 61 tiles.

Applied rules:

• Pattern recognition: Identify the constant difference between terms

• Formula creation: Develop a rule based on the pattern

• Equation solving: Use algebra to find unknown values

🧩
Pattern Rule
Fig 1: 4
Fig 2: 7
Fig 3: 10
Fig 4: 13
Fig 5: 16

Rule: 3n + 1

🔍
Finding Nth Term
3n + 1
3(15) + 1
46 tiles
Key Rules, Methods, and Formulas
GCF(a,b) × LCM(a,b) = a × b
Relationship Formula
Key definitions:

Factor: A number that divides another number evenly

Multiple: The product of a number and an integer

Prime Number: A number with exactly two factors (1 and itself)

Composite Number: A number with more than two factors

Problem-solving strategies:
  1. Understand: Read the problem carefully and identify what is being asked
  2. Plan: Decide which mathematical concept to apply (factors, multiples, or patterns)
  3. Solve: Apply the appropriate method systematically
  4. Check: Verify your answer makes sense in the context of the problem
Tip 1: To find factors, try dividing by small prime numbers first.
Tip 2: For LCM, multiply the highest powers of all prime factors.
Tip 3: For patterns, look at the differences between consecutive terms.
Tip 4: Always draw a picture or make a table for complex word problems.
Common errors: Forgetting 1 and the number itself as factors, misidentifying pattern rules, miscalculating LCM/GCF.
Key to success: Practice identifying keywords in word problems, master basic multiplication facts, and verify all answers.
Essential formulas and rules:

• Factor pairs: If a×b = n, then a and b are factors of n

• Multiples: Multiples of n are n, 2n, 3n, 4n, ...

• GCF: Greatest common factor is the largest shared factor

• LCM: Least common multiple is the smallest shared multiple

• Pattern rule: Find the relationship between position and value

Visual Problem-Solving: Patterns and Relationships
Exercise 6: Visualizing Patterns
Consider the following pattern sequences:
Sequence A: 2, 4, 6, 8, 10, ... (even numbers)
Sequence B: 3, 6, 9, 12, 15, ... (multiples of 3)
Sequence C: 1, 4, 7, 10, 13, ... (start at 1, add 3 each time)
A:2
A:4
A:6
A:8
A:10
B:3
B:6
B:9
B:12
B:15
C:1
C:4
C:7
C:10
C:13

Analysis: Different sequences follow different rules and grow at different rates.

  • Sequence A: Even numbers (rule: 2n)
  • Sequence B: Multiples of 3 (rule: 3n)
  • Sequence C: Arithmetic sequence (rule: 3n-2)

Questions & Answers

Question: I don't understand how to find all the factors of a number. Can you explain the systematic way to do it?

Answer: Great question! Here's the systematic approach to find all factors of a number:

  1. Start with 1 and the number itself (since 1 × number = the number)
  2. Check 2: Does 2 divide the number evenly? If yes, write down 2 and its partner
  3. Continue checking 3, 4, 5, etc., up to the square root of the number
  4. For each divisor found, write down both the divisor and its partner

Example with 24:

  • 1 × 24 = 24 → factors: 1, 24
  • 2 × 12 = 24 → factors: 2, 12
  • 3 × 8 = 24 → factors: 3, 8
  • 4 × 6 = 24 → factors: 4, 6
  • 5 doesn't divide 24 evenly
  • Since √24 ≈ 4.9, we stop here

All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Question: How do I know when a word problem is about factors versus multiples?

Answer: Look for these key words and situations:

Factor problems usually involve:

  • "Arranging" items into equal groups (like arranging flowers in rows)
  • "Dividing" items equally among people or containers
  • Finding "how many ways" something can be arranged
  • Keywords: arrange, divide, share, distribute equally

Multiple problems usually involve:

  • Events happening at regular intervals (buses, trains)
  • Items bought in packages of certain amounts
  • Finding when things happen "together" or "at the same time"
  • Keywords: every, together, simultaneously, schedule

Remember: Factors divide a number, multiples are created by multiplying a number.

Question: I have trouble seeing patterns in number sequences. Any tips?

Answer: Here are effective strategies for recognizing patterns:

  1. Look at the differences: Subtract consecutive terms to see if there's a consistent change
  2. Check for multiplication: See if each term is multiplied by the same number
  3. Look for geometric patterns: Shapes or visual representations often follow predictable growth
  4. Make a table: Organize position vs. value to see relationships

Example: 5, 10, 15, 20, 25...

  • Differences: 10-5=5, 15-10=5, 20-15=5, 25-20=5
  • Pattern: Add 5 each time
  • Rule: 5n (multiply position by 5)

Practice with simple patterns first, then move to more complex ones!