What do we understand by Factors?
Any number or algebraic expression that can divide another number, or another algebraic expression with no remainders is called a factor, in Mathematics. This means that the division is even, since there are no remainders.
Kinds of Factor
There are two kinds of factors:
Prime Factor - this means that if any positive integer which is greater than 1 has only two factors, namely one and that number itself, it is called a prime factor. This means that it is not completely divisible with other numbers. For example, 2 and are prime factors since they cannot be divided by anything other than 1, and themselves.
Composite Factor - this means that if any positive integer which is greater than 1 has more than two factors, it is a composite factor. Our number in question, 27, is a composite factor.
Factors are an important concept in arithmetic. Factors of 27 are simply numbers — more than two, which upon multiplication lead to the result or product of 27. In simple terms, factors are two or more numbers which upon multiplication lead to another number. Factors are also explored in algebra. Since algebra uses variables most of the time, the factors of an algebraic term are also in the form of variables. Factors of 27 are specifically unique in its nature. Understanding its factors through division, prime factorization is some key takeaways from this article.
The example below can explain this further:
(x2 + 5x + 5) is an algebraic term with two factors (x+5) and (x+ 1)
Therefore, (x+5)(x+ 1) = x2 + 5x + 5
What are the Factors of 27?
Based on what we have explained earlier, we can find the factors of 27 simply by seeing which numbers are completely divisible. Any numbers more than 1, which will yield 27 as result when multiplied together as well as leave no remainder when they are used to divide 27 will qualify as a factor.
How can the Factors of 27 be Calculated?
There are a number of ways to find out the factors of 27. Some of these are:
By Simple Division:
This means we can start to divide 27 by any number greater than 1. Whichever numbers leave no remainder will automatically qualify as a factor.
For example,
Factors of 27
By Prime Factorization:
In this method, we can find out the factors of 27 by listing out the factors which will yield 27 as a result when multiplied together.
For example, if we multiply 9 and 3, we will get 27.
Now, we have to check whether these are prime numbers or not.
3 is a prime number, so we move on to 9.
9 is not a prime number, and is a product of 3 multiplied with 3.
Hence, we can see that 27 has four factors- 1, 3, 9 and 27 itself.
A frequently asked question (FAQ) is whether a number can have a factor bigger than the number itself. The answer to that is no. Since factors are a multiplication of numbers that lead to another number, the factors are always smaller than the product.
A number itself is its greatest factor. The biggest factor for 27 is 27 itself. The value of the factor never exceeds the product value.
Pairs of factors is another approach of finding factors of a number. The pairs of factors for 27 can be derived from the single factors that we have found above: 1, 3, 9, and 27.
Use the approach below in order to understand the methodology behind a pair of two factors:
From the calculation above, it can be inferred that there are 2 pairs of two factors for 27. They are (1, 27) and (3, 9). You can observe that these pairs of factors have been derived originally from the factors obtained through division. Therefore, it is highly recommended that you use the division approach when attempting questions on factors.
Just as there are pairs of two factors, there are also pairs of three and more factors for a product. This approach is used to find the smallest factors of 27. Similar to pairs of two factors, we will be using the factors derived from the division method: 1, 3, 9, and 27.
From the calculations above, we can see that there can be a pair of three factors for 27, and they are (1, 3, 9).
FAQs on Factors of 27
1. Give some examples of prime factors.
Prime numbers are the numbers that are divisible by 1 one and the number itself. 2 is a prime number which is the product of 1 and 2 (the number itself). Remember, that 1 is not a prime number.
Some other examples of prime numbers are 3, 5, 7, etc. These numbers do not have any other factors except for 1 and the number itself.
2. Show the prime factorization method to find the factors of 27. Can factors be negative? Find negative factors of 27.
A prime number is a number which is the product of 1 and the number itself.
Factors can be positive and negative numbers. This means that a positive product can also be obtained with the multiplication of two negative factors. By simply adding the negative sign for each factor, you can get a positive product with negative factors. It can be put as follow:
Therefore the factors for 27 can also be a pair of two factors such as (-1, -27) and (-3, -9)
3. Can a number have infinite factors?
No, any number only has a given number of factors. It cannot be infinite or unlimited in any way. In our case, for example, 27 is a number with four factors only. It cannot have more than four, as we have shown, since no other numbers divide it completely. Upon multiplication of other numbers, they do not yield 27 either. Mathematically speaking, it is not possible for any number to have infinite factors. In that case, calculating these factors would be a very lengthy process.
4. How can we find the factors of a number by multiplication?
It is possible to find out the factors a given number has through multiplication. Take any number of your choice and you can find its factors using the multiplication method. List down all the ways that number can be obtained by multiplying different numbers. Keep in mind that any number will have at least two factors- the number itself, and 1. In case of prime numbers, they only have these two factors since they are not completely divisible using any other number.
5. Is it possible for 0 to be a factor for any number?
0 cannot be a factor for any number. This is because if any number is divided by 0, the result is undefined. Therefore, it cannot be said that any number is completely divisible using 0. The expression n/0 will never yield a whole number, ‘n’ being any natural number. Only if the numerator is 0 as well, this will have a definite result, i.e, 0. For all other numerators, the answer remains undefined. Hence, 1 can be a factor for all numbers, but the same cannot be said for 0.
6. Can factors of 27 be negative?
Mathematically speaking, yes. 27 can have negative factors as well, such as -1 and -27; or -3 and -9. Since two negative numbers will yield a positive number upon being multiplied together, these can be called factors of 27. However, if the question had been what are the factors of -27, the answer would be slightly different. This is because to yield a negative number, at least one of the factors would need to be negative. Therefore, 3 and -9 can be factors for -27.
7. Is it possible for numbers to have common factors?
Certainly, there can be many common factors between any two given numbers. This stands true even for prime numbers, since 1 is a common factor for every number. For example, if we consider prime numbers like 2 and 7, they have one common factor- 1. If we consider composite numbers such as 27 and 36, they have three common factors- 1, 3 and 9 . Therefore, composite numbers will usually have more factors in common with each other, than prime numbers. Even if we consider a prime and a composite factor, they will still have some common factors.
