Solved Exercises on Prime and Composite Numbers Elementary in Grade 4

Master prime and composite numbers elementary: identifying primes, finding composites, understanding divisibility, and recognizing number properties through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Prime Numbers
Exercise 1
Is 7 a prime number? Show all factors and explain your answer.
Definition:

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

Method:
  1. Start with 1 and check if it divides the number evenly
  2. Check 2, 3, 4, 5, 6, and 7 (up to the number itself)
  3. List all numbers that divide evenly
  4. Count the total factors
  5. If exactly 2 factors exist, it's prime
Number
7
Factors
1, 7
Count
2 factors
Step 1: Check divisibility by 1

7 ÷ 1 = 7 remainder 0 → 1 is a factor

Step 2: Check divisibility by 2

7 ÷ 2 = 3 remainder 1 → 2 is not a factor

Step 3: Check divisibility by 3

7 ÷ 3 = 2 remainder 1 → 3 is not a factor

Step 4: Check divisibility by 4

7 ÷ 4 = 1 remainder 3 → 4 is not a factor

Step 5: Check divisibility by 5

7 ÷ 5 = 1 remainder 2 → 5 is not a factor

Step 6: Check divisibility by 6

7 ÷ 6 = 1 remainder 1 → 6 is not a factor

Step 7: Check divisibility by 7

7 ÷ 7 = 1 remainder 0 → 7 is a factor

Factors of 7: 1, 7 (exactly 2 factors)
Final answer:

Yes, 7 is a prime number because it has exactly two factors: 1 and 7.

Applied rules:

• Prime Definition: Must have exactly two factors

• Factor Testing: Check divisibility systematically

• Counting: Only 2 factors means it's prime

2 Identifying Composite Numbers
Exercise 2
Is 8 a composite number? Find all its factors and explain why.
Definition:

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

Number
8
Factors
1, 2, 4, 8
Count
4 factors
Step 1: Check divisibility by 1

8 ÷ 1 = 8 remainder 0 → 1 is a factor

Step 2: Check divisibility by 2

8 ÷ 2 = 4 remainder 0 → 2 is a factor

Step 3: Check divisibility by 3

8 ÷ 3 = 2 remainder 2 → 3 is not a factor

Step 4: Check divisibility by 4

8 ÷ 4 = 2 remainder 0 → 4 is a factor

Step 5: Check divisibility by 5

8 ÷ 5 = 1 remainder 3 → 5 is not a factor

Step 6: Check divisibility by 6

8 ÷ 6 = 1 remainder 2 → 6 is not a factor

Step 7: Check divisibility by 7

8 ÷ 7 = 1 remainder 1 → 7 is not a factor

Step 8: Check divisibility by 8

8 ÷ 8 = 1 remainder 0 → 8 is a factor

Factors of 8: 1, 2, 4, 8 (4 factors total)
Final answer:

Yes, 8 is a composite number because it has more than two factors: 1, 2, 4, and 8.

Applied rules:

• Composite Definition: Must have more than two factors

• Factor Pairs: 1×8=8 and 2×4=8

• Counting: More than 2 factors means it's composite

3 Classifying Numbers
Exercise 3
From the list below, circle the prime numbers and underline the composite numbers: 5, 6, 9, 11, 13
Definition:

Classification: Determining if each number is prime or composite by finding its factors

Numbers
5, 6, 9, 11, 13
Factors
List each
Types
P, C, C, P, P
Step 1: Analyze 5

Factors: 1, 5 → Prime (exactly 2 factors)

Step 2: Analyze 6

Factors: 1, 2, 3, 6 → Composite (4 factors)

Step 3: Analyze 9

Factors: 1, 3, 9 → Composite (3 factors)

Step 4: Analyze 11

Factors: 1, 11 → Prime (exactly 2 factors)

Step 5: Analyze 13

Factors: 1, 13 → Prime (exactly 2 factors)

Step 6: Final classification

Prime: 5, 11, 13 | Composite: 6, 9

Prime: 5, 11, 13 | Composite: 6, 9
Final answer:

Prime numbers: 5, 11, 13 | Composite numbers: 6, 9

Applied rules:

• Systematic Analysis: Examine each number individually

• Factor Counting: Determine the exact number of factors

• Classification Rules: 2 factors = prime, more than 2 = composite

Elementary Rules and Methods
Prime: 2 Factors | Composite: >2 Factors
Elementary Classification Rule
Rule 1
Prime = exactly 2 factors
Prime Rule
Rule 2
Composite = more than 2 factors
Composite Rule
Rule 3
1 is neither prime nor composite
Special Case
Key definitions:

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

Factors: Numbers that divide evenly into another number without leaving a remainder

Elementary methodology:
  1. Check if number > 1: Prime/composite only applies to numbers greater than 1
  2. Find all factors: Systematically check divisibility from 1 up to the number
  3. Count factors: Tally the total number of factors found
  4. Classify: 2 factors = prime, more than 2 = composite
Tip 1: Even numbers (except 2) are always composite since they're divisible by 2.
Tip 2: Numbers ending in 5 (except 5) are composite since they're divisible by 5.
Tip 3: The number 2 is the only even prime number.
Tip 4: Start by checking if the number divides evenly by 1 and itself.
Common errors: Thinking 1 is prime, missing factors when counting, confusing factors with multiples.
Key concepts: Prime numbers are the building blocks of all numbers, every composite number can be broken down into primes.
Elementary rules to remember:

• 1 is neither prime nor composite

• 2 is the smallest and only even prime number

• All primes greater than 2 are odd numbers

• The first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19...

• Every composite number has at least one prime factor

Solution: Exercises 4 to 5
4 Word Problem
Exercise 4
Emma has 13 stickers. She wants to share them equally among some friends with no stickers left over. How many friends can she share with? Is 13 prime or composite?
Definition:

Word problems: Real-world situations requiring knowledge of factors and divisibility

Sticker count
13
Factors
1, 13
Sharing options
1 friend or 13 friends
Step 1: Find factors of 13

Factors of 13: 1, 13

Step 2: Interpret sharing options

Option 1: Share with 1 friend (13 ÷ 1 = 13 stickers each)

Option 2: Share with 13 friends (13 ÷ 13 = 1 sticker each)

Step 3: Determine if 13 is prime or composite

Since 13 has exactly 2 factors (1 and 13), it's prime

Step 4: Answer the question

Emma can share with 1 friend (giving 13 stickers) or 13 friends (giving 1 sticker each)

13 is prime, so Emma can share with 1 or 13 friends
Final answer:

Emma can share her 13 stickers with 1 friend (13 stickers each) or 13 friends (1 sticker each). The number 13 is prime because it has exactly two factors: 1 and 13.

Applied rules:

• Prime Application: Prime numbers have limited sharing options

• Real-world connection: Equal sharing requires understanding factors

• Verification: Confirm that sharing uses all items

5 Pattern Recognition
Exercise 5
Find all prime numbers between 10 and 20. Circle them and explain how you found them.
Definition:

Range analysis: Systematically checking numbers within a specified range

Range
10-20
Check each
11, 12, 13, 14, 15, 16, 17, 18, 19
Primes
11, 13, 17, 19
Step 1: List numbers 11-19 (excluding 10 and 20)

11, 12, 13, 14, 15, 16, 17, 18, 19

Step 2: Check 11

Factors: 1, 11 → prime

Step 3: Check 12

12 = 2 × 6 → composite

Step 4: Check 13

Factors: 1, 13 → prime

Step 5: Check 14

14 = 2 × 7 → composite

Step 6: Check 15

15 = 3 × 5 → composite

Step 7: Check 16

16 = 4 × 4 → composite

Step 8: Check 17

Factors: 1, 17 → prime

Step 9: Check 18

18 = 2 × 9 → composite

Step 10: Check 19

Factors: 1, 19 → prime

Prime numbers between 10 and 20: 11, 13, 17, 19
Final answer:

The prime numbers between 10 and 20 are: 11, 13, 17, and 19

Applied rules:

• Systematic approach: Check each number in the range individually

• Even number rule: All even numbers > 2 are composite

• Verification: Double-check each classification

Elementary Prime and Composite Concepts
Prime: 2 Factors | Composite: >2 Factors
Elementary Classification
Elementary definitions:

Prime Number: A number that can only be divided by 1 and itself. It has exactly two friends (factors) that can divide it evenly.

Composite Number: A number that has more than two friends (factors) that can divide it evenly. It's made up of smaller numbers.

Factors: Numbers that fit evenly into another number, like puzzle pieces that fit together.

Elementary approach:
  1. Start at 1: Check if 1 divides the number evenly
  2. Go up one by one: Check 2, 3, 4, etc.
  3. Count the fits: Keep track of how many numbers divide evenly
  4. Make the call: 2 fits = prime, more than 2 fits = composite
Tip 1: Even numbers (except 2) are like even teams - they can always be split in half, so they're composite.
Tip 2: The number 2 is special - it's the only even prime number.
Tip 3: Practice with small numbers first: 2, 3, 5, 7 are all prime.
Tip 4: Remember that 1 is special - it's not prime or composite.
Important facts: Prime numbers are like building blocks - everything else is made from them. They help us understand numbers better.
Real-world use: We use prime numbers to keep secrets safe on computers and phones.
Elementary rules:

• The number 1 is special - it's not prime or composite

• 2 is the smallest prime and the only even prime

• All primes greater than 2 are odd numbers

• Small prime numbers: 2, 3, 5, 7, 11, 13, 17, 19...

• Every number can be broken down into prime numbers

Questions & Answers

Question: Why is 1 not a prime number? It only goes into itself.

Answer: This is a common question that puzzles many students!

The definition of a prime number says it must have exactly two different factors: 1 and itself. Since 1 only has one factor (which is 1), it doesn't meet this rule.

Think of it like this: prime numbers are special because they can only be arranged in exactly two ways (1 row or 1 column). The number 1 can only be arranged in one way, so it's in its own special category.

So 1 is neither prime nor composite - it's just special by itself!

Question: How can I tell if a big number is prime without checking every number up to it?

Answer: Great question! You don't need to check every number!

  • Stop early: You only need to check up to the square root of the number
  • Use shortcuts: If it's even (except 2), it's composite
  • Check small primes first: Try dividing by 2, 3, 5, 7, 11

For example, to check if 49 is prime:

  • It's odd, so not divisible by 2
  • Sum of digits: 4+9=13, not divisible by 3
  • It ends in 9, not 0 or 5
  • But wait: 7 × 7 = 49, so 49 is composite!

This makes checking much faster!

Question: Are there any patterns to help me remember prime numbers?

Answer: Yes, there are helpful patterns!

  • All primes > 2 are odd (except 2, which is the only even prime)
  • Primes > 5 end in 1, 3, 7, or 9 (never 0, 2, 4, 5, 6, 8)
  • Primes become less common as numbers get bigger

The first 10 primes to remember: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

After 2 and 5, notice that all primes end in 1, 3, 7, or 9!