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Algebra

Composite numbers

Written by Prerit Jain

Composite numbers

Composite numbers

Introduction

A composite number is one that may be divided both by another number than 1 and by the number itself. Composite numbers have more than two contributing factors. Depending on how many factors they contain, numbers may be categorised. A number is a prime number if it only contains two factors: one and the number itself. The majority of numbers, known as composite numbers, include more than two components.

Types of composite numbers

There are two main types of composite numbers:

●       Odd composite numbers: All odd numbers that are not classified as prime numbers are odd composite numbers. The first few of them are 9,15,21,25,27 etc.

●       Even composite numbers: All even numbers except for 2 are composite numbers, because 2 is the only even prime number and always is a factor to the rest of the even numbers. The first few of them are 4,6,8,10,12,14 etc.

Finding composite numbers

The steps to determine if a particular number is prime or composite are as follows:

●       Find all positive integer’s factors.

●       If a number only contains the number one and itself as components, it is said to be prime.

●       A composite number is one with more than two factors.

Properties of composite numbers

●       There are more than two components in composite numbers.

●       By their factors, composite numbers can be divided equally.

●       Every composite number has itself as a factor.

●       4, which is the lowest composite number.

●       For example,15 = 3 \times 5, where 3 and 5 are prime numbers, each composite number will have at least two prime numbers as its components.

●       Additionally, composite numbers can be divided by other composite numbers.

Is 0 considered a composite number?

Any number may be written as the product of two prime numbers, according to the fundamental principle of arithmetic. It wouldn’t work if 0 were a prime number since any integer multiplied by 0 would result in 0, giving it more than 2 factors. It cannot be written as a product of two primes if it were deemed composite since doing so would require multiplying by 0, which we have assumed to be the case.

List of common composite numbers

The list of composite numbers from 1 to 100 is 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100.

Differences between prime and composite numbers

Prime numberComposite number
There are only two components since it can only be split by one and by itself.It includes more than two components (1 and itself).
The only way to write it is as the product of two numbers.The result can be expressed as the sum of two or more numbers.

Prime factorization of composite numbers

Here is a list of all the composite numbers from 1 to 50, along with a breakdown of their prime factors. Here, you can see the prime factorization of the composites. To better understand, look at the table below. This table allows you to locate composites with prime factorizations that are more than 50.

Composite numbersPrime factorization
42 \times 2
62 \times 3
82 \times 2 \times 2
93 \times 3
102 \times 5
122 \times 2 \times 3
142 \times 7
153 \times 5
162 \times 2 \times 2 \times 2
182 \times 3 \times 3
202 \times 2 \times 5
213 \times 7
222 \times 11
242 \times 2 \times 2 \times 3
255 \times 5
262 \times 13
273 \times 3 \times 3
282 \times 2 \times 7
302 \times 3 \times 5
322 \times 2 \times 2 \times 2 \times 2
333 \times 11
342 \times 17
355 \times 7
362 \times 3 \times 2 \times 3
382 \times 19
393 \times 13
402 \times 2 \times 5 \times 2
422 \times 3 \times 7
442 \times 2 \times 11
453 \times 3 \times 5
462 \times 23
482 \times 2 \times 2 \times 2 \times 3
497 \times 7
502 \times 5 \times 5

Are all composite numbers even?

No not all composite numbers are even, the numbers that may be stated as the product of two or more prime numbers are known as composite numbers.

Since we are assuming that all even numbers are composite numbers, 2 cannot be written as the sum of two or more prime numbers.

All the odd numbers that have factors of odd prime numbers are composite numbers.

Applications of composite numbers

●       In mathematics, the usage of composite numbers, or any whole integer that is not prime, is common.

●       In all areas of mathematics, the usage of composite numbers is utilised to explain shape, measure, determine how much of an object there is, and do arithmetic.

●       Numbers are used often in daily life, such as when purchasing or selling anything, paying for something, finding out the time, counting the number of runs the Indian cricket team scored during a match, counting the number of pupils seated in a classroom, calculating a student’s grades, etc.

●       A key to the code is required to decipher the encrypted data. Prime and composite integers are used in one of the most widely used encryption techniques.

●       The composite numbers, which are typically quite big, are used as codes. It is difficult to divide the composite number into its two prime elements since the number of prime numbers is unknown.

Conclusion

In this article we learnt about composite numbers and their different types. We learnt how to identify composite numbers and its properties. We also learnt about the differences between prime numbers and composite numbers, and we saw how prime factorization works and the factorization for composite numbers till 50. Composite numbers are a category of numbers we use in everyday life scenarios and are a major type of numbers.

Sample examples

Example 1: Find if 456 is a composite number

Solution 1:

The factors to 456 are: 1, 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228, 456

Since the integer has factors other than 1 and itself it is a composite number.

Example 2: What is the prime factorization of 78?

Solution 2:

The prime factors of 78 is 2 \times 3 \times 13.

Example 3: Find if the product of the first 5 prime numbers is composite.

Solution 3:

The first 5 prime numbers are:2,3,5,7,11.

The product is 2 \times 3 \times 5 \times 7 \times 11 = 2310, and we can see the factors include other factors apart from 1 and the number itself, therefore, the product is composite.

Example 4: Identify the composite numbers from the list given, 2,4,56,57,89,768,404,77.

Solution 4:

2,57,89 are prime numbers.

4,56,768,404,77 are composite numbers with factors 1, 2, 4; 1, 2, 4, 7, 8, 14, 28, 56; 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 768; 1, 2, 4, 101, 202, 404; 1, 7, 11, 77; respectively.

Example 5: What is the largest composite number between 1 and 100 with only odd numbers in its prime factorization.

Solution 5:

The largest composite number with only odd numbers in the factorization is 99 with the factorization as 3 \times 3 \times 11.

FAQs

1. What are consecutive composite numbers?

Ans: Composite numbers that follow one another in a straight line without a prime number in between are said to be consecutive composite numbers. The initial few consecutive composite numbers, for instance, may be presented as 4, 6, 8, 9, etc.

2. Does 1 classify as a composite number?

Ans: No, because 1 only has one component and does not include more than two. It is therefore neither prime nor composite.

3. What is the smallest composite number?

Ans: Other than 1 and the number itself, a composite number consists of more than two components. In this case, the number 4 meets this requirement, and no other smaller number is composite. The lowest composite number is 4, which follows the requirements.

4. Can a number be both composite and prime?

Ans: A number cannot be prime and composite at the same time. A composite number has more elements than a prime number, which only has 1 and itself as its factors. All natural numbers are either prime or composite, but not both, with the exception of 1 which is neither.

5. What is the only even number that is not composite?

Ans: 2 is the only even prime number since it has only 1 and itself as its factors.

References

Pradhan, J. B. (2019). Conceptual metaphor for teaching and learning of prime and composite numbers at primary grades. The Eurasia Proceedings of Educational and Social Sciences, 14, 78-88.

Robbins, C., & Adams, T. L. (2007). Mathematical Roots: Get Primed to the Basic Building Blocks of Numbers. MatheMatics teaching in the Middle school, 13(2), 122-127.

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Prerit Jain

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