Factors

Factors of a given number

Let's discuss the concept of a factor and identify factors of a given number. We will begin with the definition of a factor.

A factor is essentially a divisor of a given number. Dividing a number by its divisor results in a remainder equal to zero. As we have seen, the number 24 has a total of 8 factors, namely, 1, 2, 3, 4, 6, 8, 12 and 24.

Now, try this yourself: divide 24 by each of these listed factors. You will notice that upon dividing them, there will be no remainders!

Factors and negative numbers

Here is a question: can factors be negative numbers? Algebraically, we can indeed divide a number by one that is negative. Let us look at the factors of the number 6. We can factorise 6 in the following 2 ways: 1 × 6 and 2 × 3. Therefore, the factors of 6 are 1, 2, 3 and 6.

We know that the product of two negative numbers yields a positive number. Taking this into context, we find that the factors of 6 are in fact, 1, –1, 2, –2, 3, –3, 6 and –6 since –1 × (–6) and –2 × (–3) also equate to 6. However, this article will not take into account negative factors. When dealing with problems related to factors, we shall only consider the positive factors.

Characteristics of factors

There are several important properties of factors that we should familiarise ourselves with. These are listed below.

• There is a finite number of factors for a given number.

• The factor of a given number is always less than or equal to itself.

• Every number has at least two factors, that is, 1 and itself (except 0 and 1).

• We can use division and multiplication to find factors.

Identifying factors of a given number

Looking at the tip in Characteristics of Factors, there are two methods we can use to find factors: multiplication and division. Let us observe both techniques in turn.

Finding factors via multiplication

There are two steps to finding factors using multiplication.

1. Express the given whole number as a product of two whole numbers. You must consider every possible way in which this number can be written in this manner;

2. List down all the numbers that are involved in these products. These are the factors of the given number.

Finding factors via division

As for division, this follows a three-step method.

1. Identify all the whole numbers less than or equal to the given whole number.

2. Divide the given number by each of the numbers found in Step 1.

3. Write down the divisors that give the remainder of 0. These are the factors of the given number.

Applying division when determining factors can be somewhat lengthy at times, especially when dealing with larger numbers. Always be extra cautious of the algebra involved here and be sure not to miss any numbers upon dividing! However, in any case, it would indeed be safer to use the multiplication method to find the factors of a given number.

Finding the number of factors

Sometimes, it may be helpful to know the number of factors a number may have. This is particularly useful, especially when dealing with larger numbers. To do this, we need to be made aware of a new concept called prime factorisation. Taking this idea into account, we can identify the number of factors for a given number using a process called the factor tree method.

Prime factorisation

So what do prime numbers and factors have in common? Firstly, let us recall what a prime number is. A prime number is a number that has precisely two factors, 1 and the number itself. For example, the factors of 15 are 1, 3, 5, and 15. Based on this list of factors, we find that 3 and 5 are prime numbers. This brings us to the following definition.

So far, we have seen how we can represent a number as a product of two numbers. However, we can also express a number as a product of its prime factors. This is called prime factorisation.

Essentially, what we are doing here is breaking down a number in terms of its prime factors. There are two ways to determine the prime factorisation of a number.

1. Factor tree method
2. Division method

In this context, we shall only focus on the first method as it is widely used to find the number of factors for a given number. A more detailed explanation discussing this subtopic can be found in our explanation on Prime Factorisation.

The factor tree method

We shall first establish the factor tree method below.

1. Write the number at the top of the factor tree.

2. Express the number as a product of two factors branching out of the tree.

3. Further branch out each of these factors found in Step 2 as a product of two factors.

4. Repeat Step 3 until we are unable to branch out each factor. At this point, it should be written as a prime factor.

5. Finally, define the given number as a composite of its prime factors in exponent form.

Here is an example that demonstrates this process.

From here, we can identify the number of factors using the factor tree method. There are four steps to this method.

1. Find the prime factorisation of the given number using the factor tree method.

2. Express this found product of primes in its corresponding exponent form.

3. Add 1 to each exponent.

4. Multiply the numbers found in Step 3. The result yields the number of factors of the given number.

To show this, let us use our previous example.

Common factors

Let's say we are told to compare the factors between two whole numbers x and y. We find that they share a divisor that indeed divides both numbers without leaving a remainder. This is called the common factor between x and y.

To find the common factor between two (or more) numbers, we simply list down the factors of each number in separate rows and check for overlapping factors. A factor that appears in both (or more) lists is the common factor. Here are two worked examples that demonstrate this.

A more detailed discussion on this topic is presented in the article: Common Factors.

Highest common factors

In this section, we will be introduced to a direct consequence of common factors called the highest common factor (HCF). The HCF is the largest common factor of two or more whole numbers. Let us denote the HCF by the letter a. Thus, we can write the HCF of x and y by HCF(x, y) = a.

There are three ways to find the HCF of two (or more) numbers.

1. Listing factors method
2. Prime factorisation
3. Division method

Among these three techniques above, the listing factors method is the most straightforward of them all. Here, we simply list the factors of each number and find the common factors of those numbers. After that, we determine the highest common factor among this list of common factors.

For a more thorough explanation of this topic explaining all these methods, you can refer to Highest Common Factor. To show this, we shall return to the last two examples from our previous section.

Importance of factors

Finding factors of numbers is a very important tool in mathematics. It is predominantly used in mathematics to solve ratios and fractions. Factors also play a significant role in algebra, especially when dealing with equations. In this case, we can use this skill to recognise patterns found in given expressions, and reduce (or expand) such equations as a means of solving them.

Algebra and factors

As mentioned before, we can use factors in algebra when working with polynomials. For example, say we are given the expression 4x. The factors of 4x are 1, 2, 4, x, 2x, and 4x. As we shall uncover in later topics, we can use factors to perform algebraic calculations such as factoring expressions and polynomials. Here are a few more examples that demonstrate this.

In real-life situations, factoring numbers is mostly used when arranging is involved, as you have seen at the beginning of this topic. We can use factors to divide an object (or a set of items) into equal parts or split them into rows and columns.

Difference between factors and multiples

We shall end this topic by looking at the difference between factors and multiples. We will do this by means of the following table.