Solved Exercises on Factors and Divisibility in Grade 6

Master factors and divisibility: finding factors, identifying divisors, applying divisibility rules, and prime factorization through these 5 detailed exercises with comprehensive explanations.

Solution: Exercises 1 to 3
1 Finding factors of a number
Exercise 1
Find all the factors of 24. List them in ascending order and verify that their product pairs equal 24.
Definition:

Factor: A number that divides another number evenly without leaving a remainder. If a × b = n, then both a and b are factors of n.

Factor-finding method:
  1. Start with 1 and the number itself
  2. Check each integer from 2 up to the square root of the number
  3. For each divisor found, record both the divisor and its quotient
  4. Stop when you reach a repeated pair
  5. Arrange all factors in ascending order
Target number
24
Factors
1, 2, 3, 4, 6, 8, 12, 24
Verification
1×24=24, 2×12=24, 3×8=24, 4×6=24
Step 1: Start with 1 and 24

1 × 24 = 24

So 1 and 24 are factors

Step 2: Check 2

24 ÷ 2 = 12

So 2 and 12 are factors

Step 3: Check 3

24 ÷ 3 = 8

So 3 and 8 are factors

Step 4: Check 4

24 ÷ 4 = 6

So 4 and 6 are factors

Step 5: Check 5

24 ÷ 5 = 4.8 (not a whole number)

So 5 is not a factor

Step 6: Stop at 6

We already have 6 as a factor, so we stop here

Step 7: List all factors in order

1, 2, 3, 4, 6, 8, 12, 24

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Final answer:

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.

Applied rules:

• Factor definition: a divides n if n ÷ a is a whole number

• Pairing method: Factors come in pairs that multiply to the original number

• Square root limit: Only check up to √n to avoid redundancy

2 Divisibility verification
Exercise 2
Determine if 144 is divisible by 12. Show your work using division and factor verification.
Definition:

Divisibility: A number n is divisible by d if n ÷ d results in a whole number with no remainder.

Division
144 ÷ 12 = 12
Verification
12 × 12 = 144
Result
Yes, divisible
Step 1: Set up the division

144 ÷ 12

Step 2: Perform the division

144 ÷ 12 = 12

Step 3: Check for remainder

12 × 12 = 144

144 - 144 = 0 (no remainder)

Step 4: Alternative method (factorization)

144 = 12 × 12

Since 12 is a factor of 144, 144 is divisible by 12

Step 5: Conclusion

Yes, 144 is divisible by 12.

Yes, 144 is divisible by 12.
Final answer:

Yes, 144 is divisible by 12 because 144 ÷ 12 = 12 with no remainder.

Applied rules:

• Division method: Perform division and check remainder

• Factor verification: If d is a factor of n, then n is divisible by d

• Zero remainder: Divisibility occurs when remainder is 0

3 Prime factorization
Exercise 3
Find the prime factorization of 60 using a factor tree. Express the result in exponential form.
Definition:

Prime Factorization: Expressing a number as a product of prime numbers only. Every composite number has a unique prime factorization.

Number
60
Factor tree
60 = 2² × 3 × 5
Verification
4 × 3 × 5 = 60
Step 1: Start with the number

Begin with 60 at the top of the factor tree

Step 2: Break down into factors

60 = 4 × 15

Step 3: Continue breaking down

4 = 2 × 2

15 = 3 × 5

Step 4: Identify prime factors

2, 2, 3, and 5 are all prime numbers

Step 5: Express in exponential form

60 = 2² × 3¹ × 5¹

Step 6: Verify the result

2² × 3 × 5 = 4 × 3 × 5 = 60 ✓

60 = 2² × 3 × 5
Final answer:

The prime factorization of 60 is 2² × 3 × 5.

Applied rules:

• Prime factorization: Break down until only primes remain

• Factor tree: Systematic decomposition method

• Unique factorization: Fundamental theorem of arithmetic

Factors and Divisibility - Rules and Methods
\(n = a \times b \Rightarrow a \text{ and } b \text{ are factors of } n\)
Factor Definition
Divisibility by 2
Last digit even
0,2,4,6,8
Divisibility by 3
Digit sum divisible by 3
Example: 123 → 1+2+3=6
Divisibility by 5
Last digit 0 or 5
Example: 25, 40
Key definitions:

Factor: A number that divides another number evenly. If a × b = c, then a and b are factors of c.

Multiple: A number that is the product of a given number and an integer. If a × b = c, then c is a multiple of a and b.

Prime Number: A number greater than 1 that has exactly two factors: 1 and itself.

Composite Number: A number greater than 1 that has more than two factors.

Divisibility: A number n is divisible by d if n ÷ d results in a whole number with no remainder.

Complete methodology:
  1. Factor identification: Find all numbers that divide the target evenly
  2. Prime factorization: Break down numbers into prime components
  3. Divisibility testing: Apply divisibility rules to check divisibility
  4. Verification: Check results by multiplication or division
  5. Application: Use factors and divisibility in problem-solving
Tip 1: Factors come in pairs that multiply to the original number.
Tip 2: Only check divisors up to the square root of the number.
Tip 3: Use divisibility rules to quickly check if a number divides another.
Tip 4: Remember that 1 and the number itself are always factors.

Common errors: Forgetting 1 and the number itself as factors, missing factor pairs, incorrect divisibility rules.
Exam preparation: Practice with various numbers, memorize divisibility rules, understand prime factorization techniques.
Important rules to remember:

• Prime factorization is unique for each number

• Factors always come in pairs

• Every number is divisible by 1 and itself

• Divisibility by 2: last digit is even

• Divisibility by 3: sum of digits is divisible by 3

Solution: Exercises 4 to 5
4 Common factors
Exercise 4
Find all common factors of 18 and 24. Identify the greatest common factor (GCF).
Definition:

Common Factors: Numbers that divide two or more given numbers evenly. The greatest of these is the Greatest Common Factor (GCF).

Factors of 18
1, 2, 3, 6, 9, 18
Factors of 24
1, 2, 3, 4, 6, 8, 12, 24
Common factors
1, 2, 3, 6
Step 1: Find factors of 18

1 × 18 = 18

2 × 9 = 18

3 × 6 = 18

Factors of 18: 1, 2, 3, 6, 9, 18

Step 2: Find factors of 24

1 × 24 = 24

2 × 12 = 24

3 × 8 = 24

4 × 6 = 24

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

Step 3: Identify common factors

Common factors: 1, 2, 3, 6

Step 4: Find the GCF

Greatest common factor: 6

Common factors: 1, 2, 3, 6; GCF: 6
Final answer:

The common factors of 18 and 24 are 1, 2, 3, and 6. The GCF is 6.

Applied rules:

• Factor listing: Find all factors of each number

• Intersection: Identify factors that appear in both lists

• GCF: Greatest number in the common factors set

5 Divisibility rules application
Exercise 5
Use divisibility rules to determine if 1,236 is divisible by 2, 3, 4, 5, 6, and 9. Show your work for each test.
Definition:

Divisibility Rules: Quick tests to determine if one number divides another without performing the division.

Number
1236
Tests
2,3,4,6 yes; 5,9 no
Result
Divisible by 2,3,4,6
Step 1: Test for divisibility by 2

Last digit is 6 (even)

1236 is divisible by 2

Step 2: Test for divisibility by 3

Sum of digits: 1 + 2 + 3 + 6 = 12

12 is divisible by 3

1236 is divisible by 3

Step 3: Test for divisibility by 4

Last two digits: 36

36 ÷ 4 = 9

1236 is divisible by 4

Step 4: Test for divisibility by 5

Last digit is 6 (not 0 or 5)

1236 is not divisible by 5

Step 5: Test for divisibility by 6

Must be divisible by both 2 and 3

1236 is divisible by 2 and 3

1236 is divisible by 6

Step 6: Test for divisibility by 9

Sum of digits: 1 + 2 + 3 + 6 = 12

12 is not divisible by 9

1236 is not divisible by 9

Divisible by: 2, 3, 4, 6; Not divisible by: 5, 9
Final answer:

1236 is divisible by 2, 3, 4, and 6, but not by 5 or 9.

Applied rules:

• Divisibility by 2: Last digit is even

• Divisibility by 3: Sum of digits divisible by 3

• Divisibility by 4: Last two digits divisible by 4

• Divisibility by 5: Last digit is 0 or 5

• Divisibility by 6: Divisible by both 2 and 3

• Divisibility by 9: Sum of digits divisible by 9

Complete Theory: Factors and Divisibility
\(n = a \times b \Rightarrow a|n \land b|n\)
Factor Relationship
Key definitions:

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

Multiple: The result of multiplying a number by an integer.

Prime Number: A number with exactly two distinct positive divisors: 1 and itself.

Composite Number: A number with more than two distinct positive divisors.

Divisibility: The property of one number being evenly divisible by another.

Complete methodology:
  1. Factor identification: Systematically find all divisors
  2. Prime factorization: Decompose into prime components
  3. Divisibility testing: Apply appropriate rules
  4. Verification: Check results by multiplication
  5. Application: Use in problem-solving contexts
Tip 1: Prime numbers are the building blocks of all integers.
Tip 2: Use divisibility rules to save time on calculations.
Tip 3: Factor trees help visualize prime factorization.
Tip 4: The number of factors can be determined from prime factorization.

Real-world applications: Simplifying fractions, finding common denominators, cryptography, and number theory.
Advanced connections: Foundation for understanding greatest common divisors, least common multiples, and modular arithmetic.
Essential properties and rules:

• Every integer > 1 has a unique prime factorization

• Factors come in pairs whose product equals the original number

• Prime numbers have exactly 2 factors

• Composite numbers have more than 2 factors

• Divisibility by 6 requires divisibility by both 2 and 3

Visualizing Factors: Number Properties Analysis
Exercise 6: Factor Analysis
Compare the number of factors for various integers to understand their properties.

Analysis: The chart shows how the number of factors varies with different integers.

  • Prime numbers have exactly 2 factors
  • Square numbers have odd number of factors
  • Highly composite numbers have many factors

Questions & Answers

Question: What's the difference between factors and multiples?

Answer: Here are the key differences:

  • Factors: Numbers that divide a given number evenly (e.g., factors of 12: 1, 2, 3, 4, 6, 12)
  • Multiples: Products of a given number and integers (e.g., multiples of 3: 3, 6, 9, 12, 15...)
  • Relationship: If a is a factor of b, then b is a multiple of a
  • Quantity: Factors are finite; multiples are infinite

Think of factors as dividers and multiples as products.

Question: How do I remember the divisibility rules?

Answer: Here are memory aids for common divisibility rules:

  • Divisible by 2: Ends in an even number (0,2,4,6,8)
  • 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: Ends in 0 or 5
  • Divisible by 6: Divisible by both 2 and 3
  • Divisible by 9: Sum of digits is divisible by 9

Practice these regularly to build fluency.

Question: Why is 1 neither prime nor composite?

Answer: The number 1 is neither prime nor composite for specific reasons:

  • Not prime: Prime numbers must have exactly two distinct positive divisors (1 and itself). 1 only has one divisor.
  • Not composite: Composite numbers must have more than two distinct positive divisors. 1 only has one divisor.
  • Special case: 1 is called a "unit" in number theory

This exclusion is important for the Fundamental Theorem of Arithmetic.

Question: How do I find the total number of factors of a number using prime factorization?

Answer: Use the prime factorization to find the total number of factors:

If a number n has prime factorization n = p₁^a × p₂^b × p₃^c × ..., then the total number of factors is:

(a + 1) × (b + 1) × (c + 1) × ...

Example: For 60 = 2² × 3¹ × 5¹

Total factors = (2+1) × (1+1) × (1+1) = 3 × 2 × 2 = 12 factors

This works because each exponent represents the choices for that prime factor.

Question: Can negative numbers be prime?

Answer: By the standard definition, prime numbers are restricted to positive integers greater than 1:

  • Standard definition: Primes are positive integers with exactly two positive divisors
  • Reasoning: Including negatives would complicate factorization uniqueness
  • Alternative view: In more advanced mathematics, -p is considered "associate" to p

For grade school mathematics, primes are always positive.