Prime and composite numbers are important concepts in mathematics. Understanding the difference between these two types of numbers can help students better understand mathematical concepts such as multiplication, division, and factorization. In this article, we will explore what prime and composite numbers are, how to identify them, and why they are important.
Prime Numbers:
A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. In other words, it is a number that is only divisible by 1 and itself. For example, 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 are all prime numbers. The concept of prime numbers is important in mathematics and computer science because they are used in cryptography and number theory.
One important fact about prime numbers is that they can only be factored into 1 and themselves. For example, the number 7 can only be factored into 1 and 7.
The prime numbers among the first 100 positive integers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
There are a total of 25 prime numbers among the first 100 positive integers. The largest prime number among them is 97, while the smallest prime number is 2.
Twin primes :Twin Prime Numbers are the set of two numbers that have exactly one composite number between them. They can also be defined as the pair of numbers with a difference of 2.
total 8 pairs of twin primes between 1 to 100.
all pairs of twin primes have the form {6n-1, 6n+1}.
The sum of each twin prime pair except {3,5} is divisible by 12 as (6n-1) + (6n+1) = 12n.
co primes
If the only common factor of two numbers a and b is 1, then a and b are co-prime numbers. In this case, (a, b) is said to be a co-prime pair
- The Highest Common Factor (HCF) of two coprime numbers is always 1. For example, 5 and 9 are coprime numbers, there, HCF (5, 9) = 1.
- The Least Common Multiple (LCM) of two co-primes is always their product. For example, 5 and 9 are co-prime numbers. Hence, LCM (5, 9) = 45.
- 1 forms a co-prime number pair with every number.
- Two even numbers cannot be co-prime numbers as they always have 2 as the common factor.
- The sum of two co-prime numbers is always co-prime with their product. For example, 5 and 9 are co-prime numbers. Here, 5 + 9 = 14 is co-prime with 5 × 9 = 45.
- Two prime numbers are always co-prime. They have only 1 as their common factor. Consider 29 and 31. 29 has 2 prime factors, 1 and 29 only. 31 has 2 prime factors, 1 and 31 only. 29 and 31 are prime numbers. They have only one common factor 1. Thus they are co-prime. We can check any two prime numbers and get them as co-prime. For example, 2 and 3, 5 and 7, 11 and 13, and so on.
- All pairs of two consecutive numbers are co-prime numbers. Any two consecutive numbers have 1 as their common factor.
Composite Numbers:
A composite number is a positive integer that has at least one positive integer divisor other than 1 and itself. I
In other words, it is a number that can be factored into smaller positive integers. For example, 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18 are all composite numbers.
🎯key points for prime and composite numbers
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Prime numbers are positive integers greater than 1 that have no positive integer divisors other than 1 and themselves. Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
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Composite numbers are positive integers that have at least one positive integer divisor other than 1 and themselves. Examples of composite numbers include 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.
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The number 1 is neither prime nor composite because it has only one positive divisor, which is 1 itself.
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Prime numbers have exactly two positive divisors, while composite numbers have more than two positive divisors.
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All even numbers greater than 2 are composite numbers because they are divisible by 2.
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The fundamental theorem of arithmetic states that every positive integer greater than 1 can be expressed as a unique product of prime numbers.
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The number 2 is the only even prime number. All other even numbers are composite.
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To determine whether a number is prime or composite, one can try to divide it by smaller numbers. If it has a divisor other than 1 and itself, it is composite. If it has no other divisors, it is prime.
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Prime numbers are important in cryptography because they are used to create secure encryption codes.
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Composite numbers can be factored into smaller positive integers, while prime numbers can only be factored into 1 and themselves.
It is important to note that the distribution of prime numbers among the first 100 positive integers is not uniform. For example, there are 4 prime numbers between 1 and 10 (2, 3, 5, and 7), while there are only 2 prime numbers between 90 and 100 (97 and 89).
Understanding the properties of prime numbers and being able to identify them is important in many areas of mathematics, including number theory, algebra, and cryptography.
Exercises
1.What is the prime factorization of 48?
A. 2 x 3 x 4
B. 2 x 2 x 2 x 6
C. 2 x 4 x 8
D. 3 x 4 x 12
Answer: B. 2 x 2 x 2 x 6
2. Which of the following is a prime number?
A. 25 B. 27 C. 29 D. 30
Answer: C. 29
4. What is the difference between prime and composite numbers?
A. Prime numbers have exactly two factors, while composite numbers have more than two factors.
B. Prime numbers have more than two factors, while composite numbers have exactly two factors.
C. Prime numbers are odd, while composite numbers are even.
D. Prime numbers are always greater than composite numbers.
Answer: A. Prime numbers have exactly two factors, while composite numbers have more than two factors.
5.Which of the following is a composite number?
A. 41 B. 43 C. 47 D. 49
Answer: D. 49
6.What is the largest prime number among the first 30 positive integers?
A. 29 B. 31 C. 33 D. 35
Answer: B. 31
7. Which of the following is not a prime number?
A. 19 B. 21 C. 23 D. 29
Answer: B. 21
8.What is the prime factorization of 50?
A. 2 x 5 x 10
B. 2 x 5 x 25
C. 5 x 10 x 20
D. 5 x 25 x 50
Answer: B. 2 x 5 x 25
9.Which of the following is a prime number?
A. 39 B. 41 C. 42 D. 44
Answer: B. 41
10. Which of the following is a prime number?
A. 4 B. 6 C. 11 D. 16
Answer: C. 11
11.What is the smallest prime number?
A. 0 B. 1 C. 2 D. 3
Answer: C. 2
12.Which of the following statements is true?
A. All even numbers are prime numbers.
B. All odd numbers are composite numbers.
C. All prime numbers are odd.
D. All composite numbers are even.
Answer: C. All prime numbers are odd.
13.What is the prime factorization of 24?
A. 2 x 3 x 4
B. 2 x 2 x 2 x 3
C. 2 x 4 x 6
D. 3 x 4 x 8
Answer: B. 2 x 2 x 2 x 3
14 .Which of the following is a composite number?
A. 31 B. 37 C. 39 D. 41
Answer: C. 39
15.What is the largest prime number among the first 20 positive integers?
A. 17 B. 19 C. 21 D. 23
Answer: B. 19
16.The smallest 3 digit prime number is:
A.101 B.103 C.109 D.113
answer A.101
17. what is the sum of 1st 10 prime numbers?
A. 131 B. 129 C. 127 D. 135
Answer A.129
18. What is the mean of the first ten prime numbers?
A.12.9 B .13.5 C. 14 D 11.5
Answer 12.9
19.How many prime numbers are there between 1 and 50?
A) 17 B) 16 C)15 D) 20
Answer C) 15
20)How many prime numbers are there between 1 and 100?
A) 25 B)23 C)20 D) 19
Answer A) 25
