5. Prime and Composite Numbers

1. Introduction to Prime and Composite Numbers

Definition of Prime Numbers:

A prime number is a natural number greater than $1$ that has exactly two distinct factors: $1$ and the number itself i.e. number is divisible by 1 and itself only. If you try to divide by any number other than 1 and itself, then there will be some remainder. 

Examples:

• $2, 3, 5, 7, 11$ are prime numbers.

Definition of Composite Numbers:

A composite number is a natural number greater than $1$ that has more than two factors i.e. composite number can be divided by a number other than 1 and itself without leaving any remainder .
Examples:

• $4, 6, 8, 9, 12$ are composite numbers.

Special Notes:

1. 1 is neither prime nor composite.
2. $2$ is the only even prime number. All other even numbers are composite.

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2. Identifying Prime and Composite Numbers

 

To determine if a number is prime:

1. Check divisibility by $2, 3, 5, 7, \dots$ up to $\sqrt{\text{number}}$.
2. If no divisor is found, the number is prime. Otherwise, it is composite.

Example 1: Determine if $19$ is prime.

• Check divisibility:
  $19 \div 2, 19 \div 3, 19 \div 5$ → None are exact.
• Since no divisors exist other than $1$ and $19$, it is prime.

Example 2: Determine if $18$ is prime.

• Check divisibility:
  $18 \div 2 = 9$.
• Since $18$ is divisible by $2$, it is composite.

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3. Properties of Prime and Composite Numbers

 

1. Sum of two prime numbers:

   • Can be even or odd.
   • Example: $2 + 3 = 5$ (odd), $2 + 7 = 9$  (odd).

2. Product of two prime numbers:

   • Always composite.
   • Example: $3 \times 7 = 21$ .

3. Composite numbers have more factors than prime numbers.

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4. Advanced Concepts and Scenarios

 

Twin Primes:

Two prime numbers that differ by $2$.
Examples: $(3, 5), (11, 13), (17, 19)$  .

Goldbach’s Conjecture:

Every even number greater than $2$ can be expressed as the sum of two prime numbers.
Example:

• $4 = 2 + 2$,
• $6 = 3 + 3$,
• $8 = 3 + 5$.

Prime Gaps:

The difference between consecutive primes.
Example:

• Between $11$ and $13$, the gap is $2$.
• Between $13$ and $17$ the gap is $4$.

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5. Complex Scenarios with Examples

 

Example 4: Find the largest prime factor of $84$.

1. Prime factorize $84$: $$84 \div 2 = 42, \quad 42 \div 2 = 21, \quad 21 \div 3 = 7.$$
   $$84 = 2^2 \times 3 \times 7.$$
   
2. Largest prime factor: $7$.

Example 5: Express $36$ as a sum of two prime numbers.

• Possible pairs: $5 + 31$, $7 + 29$, $17 + 19$.

Answer: $36 = 5 + 31$ or $7 + 29$ or $17 + 19$.

Example 6: Prove $91$ is composite.

• Check divisibility:
  $91 \div 7 = 13$.
• Factors: $1, 7, 13, 91$.

Answer: $91$ is composite.

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7. Practice Questions with Detailed Solutions

 

Q1: Determine if $37$ is prime.

Solution:
Check divisibility by $2, 3, 5$: None are exact.
Answer: $37$ is prime.

Q2: Find the prime factorization of $72$.

Solution:

$$72 \div 2 = 36, \quad 36 \div 2 = 18, \quad 18 \div 2 = 9, \quad 9 \div 3 = 3, \quad 3 \div 3 = 1.$$
$$72 = 2^3 \times 3^2.$$

Q3: Express $50$ as a sum of two primes.

Solution:

$$50 = 3 + 47 \quad \text{or} \quad 7 + 43.$$

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8. Advanced Practice Questions with solution –

 

Q1: Identify all twin primes below $50$.

Twin primes are pairs of prime numbers that differ by $2$.

• Primes below $50$: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$
  
• Check pairs: $$(3, 5), \quad (5, 7), \quad (11, 13), \quad (17, 19), \quad (29, 31), \quad (41, 43).$$
  

Answer: Twin primes below $50$ are:
$(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43)$

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Q2: Find the smallest composite number that is the product of three distinct primes.

The smallest three distinct primes are $2, 3, 5$.
Their product:

$$2 \times 3 \times 5 = 30.$$

Answer: The smallest composite number is $30$

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Q3: Prove that $143$ is composite.

Solution: Check divisibility.

1. Start with smaller primes: $$143 \div 2 \quad \text{(not divisible, since \(143\) is odd)}.$$
   
2. Check $143 \div 3$: Sum of digits $1 + 4 + 3 = 8$ (not divisible by $3$).
3. Check $143 \div 7 = 20.428$ (not exact).
4. Check $143 \div 11 = 13$ (exact).

Factors: $1, 11, 13, 143$.

Answer: $143$ is composite.

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Q4: Express $100$ as the sum of two primes.

Solution: Test pairs of primes:

1. $100 – 3 = 97$ ($97$ is prime).
2. $100 – 11 = 89$ ($89$ is prime).

Answer: $100 = 3 + 97$  or $100 = 11 + 89$.

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Q5: Find the greatest prime factor of $630$.

Solution: Prime factorize $630$:

1. $630 \div 2 = 315$.
2. $315 \div 3 = 105$.
3. $105 \div 3 = 35$.
4. $35 \div 5 = 7$.

Prime factors: $2, 3, 5, 7$.
Greatest: $7$.

Answer: The greatest prime factor is $7$.

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Q6: Determine if $101$ is prime.

Solution: Check divisibility by primes $2, 3, 5, 7$:

• $101 \div 2$ (odd, not divisible).
• $101 \div 3 = 33.67$ (not exact).
• $101 \div 5$ (doesn’t end in $0$ or $5$).
• $101 \div 7 = 14.43$ (not exact).

No divisors exist other than $1$ and $101$.

Answer: $101$ is prime.

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Q7: Find two prime numbers whose product is $77$.

Solution: Factorize $77$:

• $77 \div 7 = 11$.

Factors: $7$ and $11$.

Answer: $7 \times 11 = 77$.

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Q8: How many prime numbers are between $50$ and $100$?

Solution: Identify primes in the range $50-100$:

$$53, 59, 61, 67, 71, 73, 79, 83, 89, 97.$$

Answer: There are $10$ prime numbers between $50$ and $100$.

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Q9: Find the LCM of $12, 15,$ and $35$ using prime factorization.

Solution:
Prime factorization:

$$12 = 2^2 \times 3, \quad 15 = 3 \times 5, \quad 35 = 5 \times 7.$$

LCM: Take the highest power of each prime.

$$LCM = 2^2 \times 3 \times 5 \times 7 = 420.$$

Answer: $LCM = 420$

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Q10: Determine the prime factors of $210$.

Solution: Factorize $210$:

1. $210 \div 2 = 105$.
2. $105 \div 3 = 35$.
3. $35 \div 5 = 7$.

Prime factors: $2, 3, 5, 7$

Answer: Prime factors are $2, 3, 5, 7$.

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Q11: Express $80$ as the product of prime numbers.

Solution: Factorize $80$:

1. $80 \div 2 = 40$, $40 \div 2 = 20$, $20 \div 2 = 10$, $10 \div 2 = 5$.

Prime factorization:

$$80 = 2^4 \times 5.$$

Answer: $80 = 2^4 \times 5$

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Q12: Prove that $231$ is composite.

Solution: Check divisibility:

1. $231 \div 3 = 77$ (exact).
2. Factors: $1, 3, 7, 11, 21, 77, 231$.

Answer: $231$ is composite.

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Q13: List all prime numbers less than $30$.

Answer: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.

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Q14: Express $144$as a product of primes.

Solution: Factorize $144$:

$$144 \div 2 = 72, \quad 72 \div 2 = 36, \quad 36 \div 2 = 18, \quad 18 \div 2 = 9, \quad 9 \div 3 = 3.$$
$$144 = 2^4 \times 3^2.$$

Answer: $144 = 2^4 \times 3^2$.

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Q15: What is the sum of all prime numbers less than $20$?

Solution: Primes less than $20$: $2, 3, 5, 7, 11, 13, 17, 19$.
Sum:

$$2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77.$$

Answer: Sum is $77$ .

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Q16: Find the prime factorization of $98$.

Solution: Factorize $98$:

$$98 \div 2 = 49, \quad 49 = 7 \times 7.$$

Prime factorization:

$$98 = 2 \times 7^2.$$

Answer: $98 = 2 \times 7^2$.

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Q17: Identify if $91$ is prime or composite.

Solution: Check divisibility:

$$91 \div 7 = 13.$$

Factors: $1, 7, 13, 91$.

Answer: $91$ is composite.

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Q18: Determine if the product of $3$ and $7$ is prime or composite.

Solution:

$$3 \times 7 = 21.$$

Since $21$ has factors $1, 3, 7, 21$, it is composite.

Answer: Composite.

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Q19: Prove $85$ is composite.

Solution:

$$85 \div 5 = 17.$$

Factors: $1, 5, 17, 85$.

Answer: $85$ is composite.

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Q20: Find two primes that sum to $28$.

Solution: Test pairs of primes:

• $28 – 5 = 23$ ($5, 23$ are primes).
• $28 – 11 = 17$ ($11, 17$ are primes).

Answer: $28 = 5 + 23$ or $28 = 11 + 17$.