Composite Numbers
Mathematics is the portal of new possibilities, with a bucket full of concepts, theories, and ideologies. There are thousands of theories and concepts discovered in Mathematics, and some are yet to be found. Mathematics is not only about theories, but it is also about their implementation in real life.
We all must once in our academic curriculum have come across the term composite numbers, which is one of the essential or integral parts in the book of pure mathematics. In mathematics, the term composite numbers can be quoted as a number having multiple factors. Contrasting with the Prime Numbers includes only two factors, i.e., one and the number itself. All-natural numbers, which are termed as prime, are composite numbers as we can divide them with multiply numbers.
For instance, we can conclude the number 8 as a composite number as it allows multiple divisions with several other numbers such as,
8 ÷ 1 = 8
8 ÷ 2 = 4
8 ÷ 4 = 2
8 ÷ 8 = 8
Therefore, the understanding of composite numbers helps us perform mathematical tasks efficiently and conveniently, saving a lot of time.
To deepen our learning of the concepts of prime numbers and composite numbers, we will dismantle different genres of some consecutive composite numbers and prime numbers.
Definition of Composite Numbers in Mathematics
Composite Numbers can be highlighted as a positive number in mathematics, which can be outputted by multiplying two smaller positive integers. Composite Numbers are those numbers that have a minimum of one divisor more aside from 1 or itself. Therefore, in simple terms, a number is composite when the number allows factorization into smaller positive integers, neither of which is one. As we know, each positive integer is prime or composite or unit 1. Therefore, composite numbers are those numbers that do not belong to the category of prime or unit 1 numbers.
For instance, if we notice the number 10. It is a composite number. It is the product of two small integers, 2 and 5. The numbers 2 and 5 are not composite numbers as both the numbers can only be divided by themselves or by 1.
We can also define every composite number as the product of two or more prime numbers. Let's take an example to clarify; the composite number 231 can be dismantled into 3 x 7 x 11; all of these are prime numbers. Again if we see the composite number 483, we can dismantle it into the product form of prime numbers, i.e., 23 x 7 x 3. This representation is termed as the order of factors. And this fact is depicted by the Fundamental Theorem of Arithmetics.
You can also grab the lesson on Composite Numbers in Hindi for free on the web.
How to Determine Composite Numbers?
Now that we are familiar with the term composite number, we must unlock the procedure to determine whether a number is prime or composite.
As we know, Composite Numbers are those positive integers that are generated by the multiplication of two small positive integers and include a minimum of one integer other than itself or 1. Therefore we can say, any even number that is greater than 2 is composite. If we see the prime numbers, they can only be divided by one or itself.
Therefore to determine whether a number is prime or composite, we must perform the following steps –
We must find all the factors of the given positive integer. As we know, any prime number has only two factors, that are one and itself. We must now notice if the positive numbers have more than one factors, they can be termed as composite numbers. For instance, if we take the positive integer 16 and now find all the factors of 16, i.e.
16 ÷ 1 = 16
16 ÷ 2 = 8
16 ÷ 4 = 4
16 ÷ 8 = 2
16 ÷ 16 = 1
Therefore, 16 has multiple factors – 1, 2, 4, 8, 16; we can conclude 16 as a composite number.
Let's take another example if we see the number 23 and find it's factors, i.e.
23 ÷ 1 = 23
23 ÷ 23 = 1
Therefore, this highlights that the number 23 has only two factors, i.e., 1 and 23 itself, so it cannot be termed as a composite number; instead, it is a prime number.
The above examples portray that any number with more than two divisors can be highlighted as a composite number. So, now let's explore more about composite numbers and their types.
Composite Numbers from 1 to 1000
Here, we present you with the complete list of composite numbers starting from the smallest composite number to 1000 –
Composite Numbers from 1 to 100 - 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.
Composite Numbers from 450 to 500 - 450, 451, 452, 453, 454, 455, 456, 458, 459, 460, 462, 464, 465, 466, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 480, 481, 482, 483, 484, 485, 486, 488, 489, 490, 492, 493, 494, 495, 496, 497, 498, 500,
Composite Numbers from 950 to 1000 - 950, 951, 952, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 968, 969, 970, 972, 973, 974, 975, 976, 978, 979, 980, 981, 982, 984, 985, 986, 987, 988, 989, 990, 992, 993, 994, 995, 996, 998, 999, 1000.
Sometimes people are confused about whether 1000 is a prime or composite number. But 1000 is a composite number as it can be expressed in the form of prime factors products. You can quickly grab the complete list of composite numbers from 1 to 1000 on the Internet.
Is 0 a Composite Number?
We know, a composite number is a positive integer n, which can be factored into smaller positive integers, keeping in mind that neither of the factors must be 1. This portrays that we can classify the positive integer into three disjoint groups –
- The unit number = (1),
- The prime numbers = (2, 3, 5, 7, 11, 13, 17,19 ...and so on),
- The composites numbers = (4, 6, 8, 9, 10, 12, 14...and so on).
To find whether the number n is composite or not, we can see its proper divisor. If the number n is small, we can perform this operation by dividing it with the primes until the square root of n. And if the number n is more extensive, we can implement one of the traditional primality tests or modern tests such as CYCLOTOMY or ECPP.
Now, if we take the case of the number 0, some may know 0 as the smallest composite number. But no, the number 0 comes neither in the category of composite number not in prime numbers. As Zero cannot be expressed in the form of products of two factors, neither is itself. Therefore we can't highlight Zero as a prime or composite number.
Even Composite Numbers
Even Composite numbers can be outlined as all the even integers that are not prime: those numbers with more than two factors.
List of Even Composite numbers from 1 to 200 – 4, 6, 8,10, 12, 14, 16, 18, 20,22, 24, 25, 26, 28, 30, 32,34,36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62,64, 65, 68, 70, 72, 74, 76,78, 80,82, 84, 86, 88, 90, 92, 94, 96, 98,100, 102, 104, 106, 108, 110,112, 114, 116, 118, 120,122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 161, 162, 164, 165, 166, 168, 169, 170, 172, 174, 176, 178, 180, 182, 184,186, 188, 190, 192, 194, 196, 198, 200
Let’s consider the numbers 1, 2, 3, 4, 9, 10, 11, 12 and 15.
Here, we can say the numbers 4, 10, and 12 are the even composite numbers as all of these numbers contain odd divisors and satisfies the conditions of being a composite number. Therefore, any positive even integer that offers multiple divisors, other than one and itself, can be quoted as an even composite number.
Odd Composite Numbers
A composite number is highlighted as a number that includes a minimum of one factor, other than one or itself. According to this, an Odd Composite number can be highlighted as all no-prime positive integers, which possess at least one factor other than one or itself.
List of Odd Composite Numbers from 1 to 200 - 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93,95, 99, 105, 111, 115, 117, 119, 121, 123, 124, 125, 129, 133, 135, 141, 143, 145, 147, 153, 154, 155, 159, 161, 165, 169, 171, 175, 177,183, 185,187,189, 195.
So, here we present you the list of odd composite numbers from 1 to 200. The number Zero is not included in either of the odd or even composite numbers as the number Zero is not a composite number because it does not have any factors.
In a few instances, the term composite number may be used in a broader sense to incorporate negative integers also. We can say a negative integer to be composite if highlighted as negative of a composite number.
Difference between Prime and Composite Numbers
We present you with a clear view of Prime Numbers and Composite Numbers –
So, what are Prime Numbers and Composite Numbers?
A prime number is a number that contains two factors, i.e., one and the number itself. While a composite number is a number with more than one factor, which depicts that other than dividing it by one and the number itself, we can divide it by at least one or more than one integer.
Therefore, the prime number can be portrayed as a number with exactly two factors, which implies it can only be divided by the number itself and by 1. While the number 1 is not considered as Prime number.
Some examples of Prime Number – 11 are a prime number as it can only be divided by 1 or 11.
i.e., 11 ÷ 1 = 11
11 ÷ 11 = 1
Some of the other prime numbers are – 2, 5, 7, 13, etc.
While a Composite number is outlined as a number with at least one factor, other than one and the number itself, the number '1' is not considered a composite number.
Some examples of composite numbers – 10 can be considered as a composite number as other than one and the number itself, it has multiple factors, such as –
10 ÷ 1 = 10
10 ÷ 2 = 5
10 ÷ 5 = 2
10 ÷ 10 = 1
I hope you are now quite familiar with the terms prime and composite.
Smallest Composite Numbers
The number 4 is the smallest composite number, which is the smallest even composite number also.
But why? You may wonder why 2 is not a composite number. The answer is two. A composite number is quite simple; 2 is not a composite number as it is a prime number because it has exactly two factors, i.e., it can be only divided by only one and the number 2 itself.
While 9 is the smallest odd composite number. Sounds weird, right? Why not, 3? The answer is again quite simple because 3 is a prime number with two factors: 1 and 3 itself. Simultaneously, the smallest two-digit composite number is 10, as it has multiple divisors such as 1, 2, 5, 10. Therefore it works according to the guidelines of being a composite number.
Some may also wonder why one is not the smallest composite number. As the sole divisor of 1 is the number 1 itself. Therefore it's not a composite number. The integers 2 and 3 are prime numbers as mentioned above. Thus, the crown of the smallest composite number is presented to number 4, as it has multiple divisors such as 1, 2, 4. So, the number satisfies the principle of being a composite number.
Prime Factorisation of Composite Numbers
Prime Factorisation of Composite Numbers is the method of breaking down the composite number into prime numbers. When we multiply all of these prime numbers, we can get the original composite number. Each number has a unique prime factorization. In simple words, the break down of composite numbers into factorable, residual parts is known as Prime Factorisation of Composite Numbers.
Let's dive in to explore how to find the Prime Factorisation of Composite Numbers –
- Step 1 – First position the prime factor on the left and the output or the quotient under it.
- Step 2 – Now, you must divide the result or the quotient with the prime number, keep following this process until the button number outputs in prime number – Now you can end the process. It does not depend on the order of execution, as the list of prime numbers will be similar, no matter the order of execution.
- Step 3 – Now use the prime factorization using the number on the left side – Properly observe the numbers down the left side of the problem and the output at the button. They are identical to remainder in a division problem, but in this case, they are the prime numbers.
Frequently Asked Questions
Q1: Define Composite Number.
Ans – The term Composite number can be quoted as a natural number or a positive integer, which includes more than two factors. That is, it has at least one factor other than one or the number itself.
For instance, we can say 16 is a composite number as it has more than two divisions such as –
16 ÷ 1 = 16
16 ÷ 2 = 8
16 ÷ 4 = 4
16 ÷ 8 = 2
16 ÷ 16 = 1
Hence, 16 is a composite number with 1, 2, 4, 8, 16 as it’s divisors.
Q2: Is two a composite number?
Ans – A composite number can be defined as a number with a minimum of one divisor other than one and the number itself. So, to answer the question – is two a composite number? If we take the case of number 2, it has exactly two divisors: – 2 ÷ 1 = 2 and 2 ÷ 2 = 1, which highlights it as a prime number, which has exactly two factors a composite number.
Q3: What are the composite numbers from 100 to 200?
Ans – The list of composite numbers from 100 to 300 are –
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, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150, 152, 153, 154, 155, 156, 158, 159, 160, 161, 162, 164, 165, 166, 168, 169, 170, 171, 172, 174, 175, 176, 177, 178, 180, 182, 183, 184, 185, 186, 187, 188, 189, 190, 192, 194, 195, 196, 198, 200.
Q4: Is nine a composite or prime number?
Ans – To verify whether 9 is a prime or composite number, we must first know what prime and composite numbers are.
A composite number is outlined as a number that contains a minimum of one divisor other than one and the number itself. Simultaneously, a prime number can be illustrated as a number, including precisely two divisions: 1 and the number itself.
So, in the case of 9, it has multiple factors such as –
9 ÷ 1 = 9
9 ÷ 3 = 3
9 ÷ 9 = 1
Therefore, the number 9 is composite and not a prime number.
Q5: Is 17 a Prime Number or Composite Number?
Ans - To check whether 17 is a prime or composite number, we must first highlight what is prime and composite.
A composite number is portrayed as a number with a minimum of one factor other than the number itself. In contrast, a prime number can be represented as a number having exactly two divisions: one and the number itself.
So, in the case of 17, it has only two factors, that is –
17 ÷ 1 = 17
17 ÷ 17 = 1
Therefore, the number 17 can be concluded as a prime number, having exactly two divisors and not a composite number.
Q6: Why is four the smallest composite number?
Ans – One may think 2 to be the smallest composite number. But in reality, it is not as it has only two factors – 1 and the number 2 itself. Therefore it is a prime number. And if we take the case of number 3, it is also a prime number with 1 and 3 as the two factors. Therefore, the smallest composite number is taken as 4, which has multiple factors like –
4 ÷ 1 = 4
4 ÷ 2 = 2
4 ÷ 4 = 1
So, with 1, 2, 4 as the factors, 4 gets the position of the smallest composite number.
Q7: Why are 0 and 1 are composite numbers?
Ans – The definition of Composite Numbers says a number is composite if the number has the minimum of one factor, other than 1, and the number itself.
So, if we take the number 0, it is neither a prime number nor a composite number as it has no factors or divisors.
And in the case of the number 1, it is not included in the list of composite numbers as the sole divisor of the number 1 is one itself, which makes it unworthy of becoming a composite number.
Q8: What is a co-prime number?
Ans – We can portray two integers, suppose x and y to be co-prime or mutually prime if 1 is the only positive divisor of the numbers, which divides them evenly.
Q9: What is the difference between Odd and Even composite numbers?
Ans – The odd composite numbers are those natural odd numbers which has at least one factor, other than 1 or the number itself. Some of the odd composite numbers are - 9, 15, 21, 25,27, 33, 35, 39, 45, 49,51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93,95, 99, 105, 111, 115, 117, 119, 121, 123, 124, 125, 129, 133, 135, 141, 143, 145, 147.
While even composite numbers are those even natural numbers which includes at least one factor, other than 1 or the number itself. Some of the odd composite numbers are - 4, 6, 8,10, 12, 14, 16, 18, 20,22, 24, 25, 26, 28, 30, 32,34,36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62,64, 65, 68, 70, 72, 74, 76,78, 80,82, 84, 86, 88, 90, 92, 94, 96, 98,100, 102, 104, 106, 108, 110,112, 114, 116, 118, 120,122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150.