Factors of 60

Factors of 60 are those numbers that divide 60 completely without leaving any remainder. There are 12 factors of 60 among which 60 is the biggest factor and 2, 3 and 5 are its prime factors. The prime factorization of 60 can be done by multiplying all its prime factors such that the product is 60. Let us learn about all factors of 60, the prime factorization of 60, and the factor tree of 60 in this article.

The factors of 60 can be listed as 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. According to the definition of factors, the factors of 60 are those numbers that divide 60 without leaving any remainder. In other words, if two numbers are multiplied and the product is 60, then the numbers are the factors of 60. It means that 60 is completely divisible by all these numbers. Apart from these, 60 also has negative factors that can be listed as, -1, -2, -3, -4, -5, -6, -10, -12, -15, -20, -30, and -60.. For negative factors, we need to multiply a negative factor by a negative factor, like, (-20) × (-3) = 60.

Factorization of a number means writing the number as a product of its factors. The multiplication method is the most commonly used method to find the factors of a number. Let us find the factors of 60 using multiplication.

Factors of 60 using Multiplication

Let us find the factors of 60 using the multiplication method using the following steps.

• Step 1: In order to find the factors of 60 using multiplication, we need to check what pairs of numbers multiply to get 60, so we need to divide 60 by natural numbers starting from 1 and go on till 9. We need to make a note of those numbers that divide 60 completely.
• Step 2: The numbers that completely divide 60 are known as its factors. We write that particular number along with its pair and make a list as shown in the figure given above. As we check and list all the numbers up to 9, we automatically get the other pair factor along with it. For example, starting from 1, we write 1 × 60 = 60, and 2 × 30 = 60 and so on. Here, (1, 60) forms the first pair, (2, 30) forms the second pair and the list goes on as shown. So, as we write 1 as the factor of 60, we get the other factor as 60; and as we write 2 as the factor of 60, we get 30 as the other factor. Like this, we get all the factors.
• Step 3: After the list is noted, we get all the factors of 60 starting from 1 up there, coming down and then we go up again up to 60. This gives us a complete list of all the factors of 60 as shown in the figure given above.

Therefore, the factors of 60 can be listed as 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Now, let us learn about the prime factorization of 60.

Prime factorization is a way of expressing a number as a product of its prime factors. The prime factors of a number are those factors that are prime numbers. The prime factorization of 60 can be done using the following steps. Observe the figure given below to understand the prime factorization of 60.

• Step 1: The first step is to divide the number 60 with its smallest prime factor. We know that a prime factor is a prime number which is a factor of the given number. So, with the help of divisibility rules, we find out the smallest factor of the given number. Here, we get 2. Therefore, 2 is the smallest prime factor of 60. So, 60 ÷ 2 = 30
• Step 2: We need to repeatedly divide the quotient by 2 until we get a number that is no more divisible by 2. So, we divide 30 again by 2 which is 30 ÷ 2 = 15
• Step 3: Now, 15 is not completely divisible by 2, so, we proceed with the next prime factor of 60, which is 3. So we will divide 15 by 3, that is, 15 ÷ 3 = 5
• Step 5: Since 5 is a prime number it will be divided by 5 and we will get 1 as the quotient.
• Step 6: We need not proceed further as we have obtained 1 as our quotient.
• Step 7: Therefore, the prime factorization of 60 is expressed as 2 × 2 × 3 × 5 = 2² × 3 × 5; where 2, 3, and 5 are prime numbers and the prime factors of 60.

Therefore, the prime factors of 60 are 2, 3, and 5 and the Prime factorization of 60 = 2 × 2 × 3 × 5

We can also find the prime factors of 60 using a factor tree. The factor tree of 60 can be drawn by factorizing 60 until we reach its prime factors. These factors are split and written in the form of the branches of a tree. The final factors are circled and are considered to be the prime factors of the 60. Let us find the prime factors of 60 using the following steps and the factor tree given below.

• Step 1: Split 60 into two factors. Let us take 2 and 30.
• Step 2: Observe these factors to see if they are prime or not.
• Step 3: Since 2 is a prime number we circle it as one of the prime factors of 60. We move on to 30, which is a composite number and further split it into more factors. In other words, we repeat the process of factorizing the composite numbers and splitting it into branches until we reach a prime number.
• Step 4: After factorizing 30, we get 2 and 15. So, we circle 2 because it is a prime number and we split 15 into 3 and 5. At this stage, we are left with prime numbers, 2, 3, and 5. We circle them since we know that they cannot be factorized further. This is the end of the factor tree.
• Step 5: Therefore, the prime factors of 60 = 2 × 2 × 3 × 5

Note: It should be noted that there can be different factor trees of 60. For example, we can start by splitting 60 into 3 and 20. Since 3 is already a prime number, we circle it and then we split 20 further into 2 and 10. Again, we get 2 as the prime number, so we circle it and split 10 into 2 and 5. Finally, we can observe the same prime factors, that is, 60 = 2 × 2 × 3 × 5

The factors of 60 can be written in pairs. This means that the product of the pair factors of 60 is always 60. The factors of 60 in pairs can be written as shown in the table given below:

It is possible to have negative pair factors as well because the product of two negative numbers also gives a positive number. Let us have a look at the negative pair factors of 60.

The following points explain some features of the pair factors of 60.

• The pair factors of the number 60 are whole numbers in pairs that are multiplied to get the original number, i.e., 60.
• Pair factors could be either positive or negative but they cannot be fractions or decimal numbers.
• The positive pair factors of 60 are as follows: (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), and (6, 10). The negative pair factors of 60 are (-1, -60), (-2, -30), (-3, -20), (-4, -15), (-5, -12), and (-6, -10)

Important Notes

• 60 is a composite number because it has factors other than 1 and itself.
• Factors of 60 are those numbers that divide 60 completely without leaving any remainder.
• 60 has a total of 12 factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
• There is a trick to calculate the total number of factors of a number. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. We get the prime factorizations of 60 as 2² × 3 × 5. Just add one (1) to the exponents 2, 1 and 1 individually and multiply their sums. (2 + 1) × (1 + 1) × (1 + 1) = 3 × 2 × 2 = 12. This means 60 has 12 factors in all.

Points to remember

Let us recollect the list of the factors, the negative factors, and the prime factors of 60.

• Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60
• Negative Factors of 60: -1, -2, -3, -4, -5, -6, -10, -12, -15, -20, -30 and -60
• Prime Factors of 60: 2, 3, 5
• Prime Factorization of 60: 2 × 2 × 3 × 5 = 2² × 3 × 5

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