Prime and Composite Numbers with Perfect Number: Definition, Properties, Examples

Prime and Composite Numbers with Perfect Number: In number theory, positive integer numbers equal to the sum of their positive divisors are perfect numbers, excluding the number itself. Prime numbers are the numbers having only two factors, namely \(1\) and the number itself.

At the same time, composite numbers are numbers that are having more than two factors. This means \(1\) is neither a prime nor a composite number. It is a unique number. In this article, let us study some facts and what are these different types of sets of numbers and how they are found. Also, we will solve examples using these numbers.

What are Perfect Numbers?

The positive integers equal to the sum of their divisors except for the number itself are known as perfect numbers. In other words, perfect numbers are the positive integers that are the sum of their divisors. The smallest perfect number is \(6,\) because it is the sum of its proper divisor \(1, 2,\) and \(3.\)

List of Perfect Numbers

The perfect numbers given in the table are the first four perfect numbers known to us since ancient times. Let us look into their divisor and their sum given in the table given below:

How to Find a Perfect Number?

Perfect numbers are less known to students as compared to other types of numbers. Let us learn how to generate the perfect numbers easily.

1. To find perfect numbers, there are no fixed propositions.
2. The formula for obtaining the four known perfect numbers \({2^{p – 1}}\left( {{2^p} – 1} \right)\) where \(p\) is a prime number.
3. Prime numbers of the form \({2^p} – 1\) is known as Mersenne primes.

Some are given below.

Proposition 1

Numbers set out in double proposition are as many as those beginning from the unit until the sum becomes prime, then the sum and last number make a perfect number. Double proposition means each number in a pattern is twice the preceding number. For example, \(1+2+4=7\) is a prime number.

According to Proposition \(1,\) sum \(×\) last number \(=\) perfect number

\(⇒7×4=28\)

And \(28\) is a perfect number.

Proposition 2

If \(N\) is a Mersenne prime, then \(\frac{{N(N + 1)}}{2}\)  gives the perfect number. A prime number that is one less than the power of \(2\) is Mersenne prime. For example, let us take \(N\) as \(31,\) which is \(1\) less than \({2^4}.\) Then, \(\frac{{N(N + 1)}}{2} = \frac{{31(31 + 1)}}{2} = \frac{{31 \times 32}}{2} = 496\)

And \(496\) is a perfect number.

Prime Numbers

Let’s learn everything about prime numbers now:

Prime Numbers Definition

Prime numbers may be defined as numbers with exactly two factors, i.e., \(1\) and the number itself. Prime numbers are greater than \(1.\) We can only divide the given number into groups with an equal number of objects if it can be factorized as a product of two numbers. \(7\) is unable to be divided into groups of equal numbers. This is because \(7\) can be factorized as follow:

1. \(7×1=7\)
2. \(1×7=7\)

\(7\) has two factors only; they are \(1\) and \(7.\) Thus, \(7\) is a prime number as it cannot be divided into groups of equal numbers.

List of Prime Numbers: There are \(25\) prime numbers from \(1\) to \(100.\) They are \(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\)

Properties of Prime Numbers

The following are some of the properties of prime numbers:

1. A prime number is always greater than \(1.\)
2. It has two only two factors that are \(1\) and the number itself.
3. The only even prime number is \(2.\)
4. Two primes are co-prime to each other always.
5. Each number can be expressed as the product of primes.

Prime Number Chart

A chart that shows the list of prime numbers in a systematic order is known as a prime number chart. Below is the chart showing prime numbers from \(1\) to \(100\)

Is There a Pattern in Prime Numbers?

Prime numbers between two numbers can be discovered by following a certain pattern. Observe the following figure with prime numbers encircled, and the numbers divisible by these prime numbers have been struck off. The pattern can be followed until the square root of the larger number \((100)\) is reached.

Composite Numbers

Let’s learn everything about composite numbers:

What are Composite Numbers?

Natural numbers having more than two factors are known as composite numbers. In other words, numbers are divisible by a number other than \(1,\) and the number itself is the composite number. 

Examples: \(4, 6, 8, 9, 10….\)are some of the composite numbers

Properties of Composite Numbers

A composite number is a positive integer formed by multiplying two smaller positive integers. Few properties of composite numbers are:

1. All composite numbers are divisible by smaller numbers, which can be prime or composite, in the same way.
2. Every composite number consists of at least two prime numbers.

The figure below explains the composite number \(72\) and the concept behind it. Let us have a look into it.

The above figure shows that by multiplying these positive integers, we get a composite number.

Finding Composite Numbers

To find a composite number, we have to find the factors of the given number. If the number has more than two factors, then it is composite. By performing the divisibility test, we can figure out a composite number easily. A number is evenly divided by another number is known as divisibility. The divisibility test helps us find out whether the number is a prime or composite number.

In this method, we divide the number by the common factors such as \(2, 3, 5, 7, 11,\) and \(13.\) They the number is even, we check with \(2,\) and if the number ends with \(0\) and \(5,\) we check with \(5.\) If the number s not divisible by any of the common factors, then the number is prime. If it does, then it is a composite number. For instance, \(68\) is divisible by \(2,\) which means it has factors other than \(1\) and \(68.\) So \(68\) is a composite number.

Types of Composite Numbers

There are two main types of composite numbers, namely odd numbers, and even composite numbers. Let us discuss each one of them:

Odd Composite Numbers

Odd numbers that are not prime are known as odd composite numbers. 

Example: \(9, 15, 21, 25, 27\) are odd composite numbers.

In the set of numbers \(1, 2, 3, 4, 9, 10, 11, 12, 15,\) the odd composites are \(9\) and \(15.\)

Even Composite Numbers

Even numbers that are not prime are known as even composite numbers.

Example: \(4, 6, 8, 10, 12, 14, 16\) are even composite numbers.

In the set of numbers \(1, 2, 3, 4, 9, 10, 11, 12, 15,\) the even composite numbers are \(4, 10,\) and \(12\)

Smallest Composite Number

We learned what composite numbers are. Let us see which is the smallest composite number. Starting from \(1, 2, 3, 4,5, 6,…\) so on. \(1\) is not a composite number as its only divisor is \(1. 2\) and \(3\) are prime numbers. \(4\) has divisors \(1, 2,\) and \(4. 4\) satisfies the condition of a composite number. Thus, the lowest composite number is \(4.\)

Solved Examples on Prime and Composite Numbers with Perfect Numbers

Q.1. Determine whether \(486\) is a composite number or not?
Ans: The given number is \(486.\) Its factors or divisors are\(-1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243,\) and \(486.\) This indicates that it has a divisor other than \(1\) and itself.
\(∴ 486\) is a composite number.

Q.2. Workout the following riddle: “I lie between \(30\) and \(40;\) I have only two prime factors. On adding my prime factors, we get \(5;\) Who am I?” A prime number or a composite number.
Ans: The sum of the prime factors of this unknown number is \(5.\) We know that the prime numbers whose sum is \(5\) are \(2\) and \(3.\) We need to find a number with \(2\) and \(3\) as its only prime factors and lies between \(30\) and \(40.\) Let us see the prime factorization of \(36:36=2×2×3×3.\) So, \(36\) satisfies the given conditions.
\(∴ 36\) is the required number. It is a composite number.

Q.3. Which of the two numbers is a prime number, \(13\) or \(15\)?
Ans: The number \(15\) has more than two factors: \(1, 3, 5,\) and \(15.\) Hence, it is a composite number. While \(13\) has only two factors: \(1\) and \(13.\) Hence, it is a prime number. Therefore, \(13\) is a prime number.

Q.4. Why is \(20\) not a prime number?
Ans: The factors of \(20\) are \(1, 2, 4, 5, 10,\) and \(20.\) Thus, \(20\) has more than two factors. As the number of factors of \(20\) is more than two numbers, it is a composite number (\(20\) is not a prime number).

Q.5. Is \(28\) a perfect number?
Ans: The proper factors of \(28\) are \(1, 2, 4, 7,\) and \(14.\) The sum of proper factors is \(28.\) As per the definition of perfect numbers, \(28\) is a perfect number. Therefore, \(28\) is a perfect number.

Summary

Perfect numbers, prime numbers, and composite numbers were the topics of this article. All three types of numbers had their definitions taught to us. Their properties, as well as the prerequisites for perfect, prime, and composite numbers, are explored. We have gone through the perfect numbers, prime numbers, and composite numbers, as well as how to find them. Students can use the examples to obtain a better understanding of these unusual numbers.

FAQs on Prime and Composite Numbers with Perfect Numbers

Q.1. Is the perfect number and prime number the same?
Ans: No, perfect and prime numbers are not the same.

Q.2. Why are there no perfect prime numbers?
Ans: The known perfect numbers are \(6, 28, 496,\) and \(8128,\) in which none are having only \(1\) and the number itself as their factors. All the perfect numbers have more than two factors, and hence they are not prime.

Q.3. Is it a perfect square composite?
Ans: A perfect square (square number) is always a composite number because the perfect square will always have more than two factors. Examples \(-10601, 11311, 12421.\)

Q.4. Is 132 a composite number?
Ans: Yes, \(132\) is divisible by \(2,\) which means it has factors other than \(1\) and \(132.\) So \(132\) is a composite number.

Q.5. How many perfect numbers are there?
Ans: \(51,\) as of \(2019,\) there are \(51\) known perfect numbers in total.

Study About Composite Numbers Concept Here

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