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Factors of 600: Prime Factorization, Methods, and Examples

The factors of 600 are the numbers that can divide the number 600 evenly or exactly without leaving any remainder

For getting the pair factors of 600, multiply any two numbers which results in 600 as the product. The numbers whose product gives the result 600 are called factors of number 600. The set of these two numbers is also called one of the factor pairs. 600 is an even composite number and has 24 factors in total.

In this complete guide, let us explore the factors of 600, and how to find them using different methods which are prime factorization and division methods.

What Are the Factors of 600?

The factors of 600 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, and 600.

All the above numbers are perfect divisors of 600. When 600 is divided by these numbers, it is divided completely with no remainder.

Also, note that 1 and the number itself are always factors of every number. So, 1 and 600 are factors of 600.

How To Calculate the Factors of 600?

To find the factors of 600, start dividing 600 by the smallest natural number that divides 600 exactly.

Divide 600 by the smallest natural number i.e., 1.

\[\dfrac{600}{1}=600 , r = 0 \]

As it has completely divided 600 without any remainder, so 1 is a factor of 600.

Now, divide 600 by the smallest even prime number i.e., 2

\[\dfrac{600}{2}=300 , r = 0 \]

As it has again divided 600 completely, so 2 is also a factor of 600.

Again divide 600 by the smallest odd prime number i.e., 3

\[\dfrac{600}{3}=200\]

As 3 has divided 600 exactly. So, 3 is too a factor of 600.

For getting more factors, divided 600 by natural numbers that exactly divide 600 and leave zero remainders as shown below:

\[\dfrac{600}{4}=150\]

\[\dfrac{600}{5}=120\]

\[\dfrac{600}{6}=100\]

\[\dfrac{600}{8}=75\]

\[\dfrac{600}{10}=60\]

\[\dfrac{600}{12}=50\]

\[\dfrac{600}{15}=40\]

\[\dfrac{600}{20}=30\]

\[\dfrac{600}{24}=25\]

\[\dfrac{600}{25}=24\]

\[\dfrac{600}{30}=20\]

\[\dfrac{600}{40}=15\]

\[\dfrac{600}{50}=12\]

\[\dfrac{600}{60}=10\]

\[\dfrac{600}{75}=8\]

\[\dfrac{600}{100}=6\]

\[\dfrac{600}{120}=9\]

\[\dfrac{600}{150}=4\]

\[\dfrac{600}{200}=3\]

\[\dfrac{600}{300}=2\]

\[\dfrac{600}{600}=1\]

Hence, all the above numbers exactly divide 600 without leaving any remainder, so all the above numbers are factors of 600.

Factors of 600 by Prime Factorization

To find factors of 600 by the prime factorization method, divide 600 by the smallest prime number which divides 600 exactly without any remainder. Then the quotient is again divided by the smallest prime number and the procedure continues until we get the quotient as 1.

Following is the method to calculate factors of 600 by prime factorization.

First, divide 600 by the smallest prime number which is 2.

\[\dfrac{600}{2}=300\]

The quotient 300 is a composite number and can further be divided by 2.

\[\dfrac{300}{2}=150\]

Again 150 is a composite number that can be further divided by 2.

\[\dfrac{150}{2}=75\]

Now 75 again can be divided further by 3.

\[\dfrac{75}{3}=25\]

25 further can be divided by 5.

\[\dfrac{25}{5}=5\]

5 can be further divided by 5.

\[\dfrac{5}{5}=1\]

The quotient 1 cannot be further divided.

Therefore, the prime factorization of 600 can be stated as:

Prime Factorization = 2 × 2 × 2 × 3 × 5 × 5

Prime Factorization of 900 can also be written as:

\[600 = 2^3 \times 3\times 5^2 \]

The Prime factorization of 600 is also shown in Figure 1 below:

Figure 1

Factor Tree of 600

A factor tree is a way to express the factors of a number, specifically the prime factorization of a number in which each branch in the tree splits into factors.

Once the factor at the end of the branch is a prime number, and the other is a composite number. Divide the composite number again unless the only two factors remain, that is itself and 1 so the branch stops.

If we write 600 into multiples, it would be 600 = 2 × 300

On dividing 300 into its multiples, it would be 300 = 2 × 150

Dividing further 150 into its multiples. It would result in 150 = 2 × 75 

On further dividing 75 into its multiple factors, it would be 75 = 3 × 25 

By splitting 25 further and writing its multiples, it would be 25 = 5 × 5

By dividing 5 further into its multiples, it would be  5 = 5 × 1

Altogether expressing the number in terms of prime factors would be:

2 × 2 × 2 × 3 × 5 × 5

The factor tree of 600 is shown in figure 2 as:

Figure 2

Factors of 600 in Pairs

A set of two natural numbers, whose product gives us the number 600 are called factors of 600 in pairs.

Pair Factors are a pair of numbers that are multiplied by each other and give the result of 600 itself. Following are the pair factors of 600.

\[1 \times 600 = 600\]

\[2 \times 300 = 600\]

\[3 \times 200 = 600\]

\[4 \times 150 = 600\]

\[5 \times 120 = 600\]

\[6 \times 100 = 600\]

\[8 \times 75 = 600\]

\[10 \times 60 = 600\]

\[12 \times 50 = 600\]

\[15 \times 40 = 600\]

\[20 \times 30 = 600\]

\[24 \times 25 = 600\]

As there are 24 factors of 600. So, these factors can be written in pairs as follows:

\[(1, 600)\]

\[(2, 300)\]

\[(3, 200)\]

\[(4, 150)\]

\[(5, 120)\]

\[(6, 100)\]

\[(8, 75)\]

\[(10, 60)\]

\[(12, 50)\]

\[(15, 40)\]

\[(20, 30)\]

\[(24, 25)\]

600 also can have two negative numbers as pair factors. For example:

\[(-12) \times (-50)=600\]

\[(-6) \times (-100)=600\]

\[(-3) \times (-200)=600\]

Hence, the following are some examples of negative pair factors of 600:

\[(-12, -50)\]

\[(-6, -100)\]

\[(-3, -200)\]

So, it can be derived that the product of all factors of 600 in its negative form, gives the result 600. So, all are called negative pair factors of 600.

Important Facts about 600

  1. 600 is a composite number.
  2. 600 is also an even number.
  3. 600 has only 3 prime factors.
  4. 600 has 24 divisors.
  5. 600 has 24 positive factors and 24 negative factors.
  6. 300 is the biggest factor of 600 excluding 600 itself.

Factors of 600 Solved Examples

Example 1

Dennis has been given 4 sets of pair factors of 600 and has been asked to pick a pair factor with one prime and one composite number. Please help him to pick from the given pair factors options.

  1. (3, 200)
  2. (8, 75)
  3. (12, 50)
  4. (24, 25)

Solution

The factor pair consisting of one prime number and one composite number is (3, 200)

Example 2

Which of the following statement is False about factors of 600?

  1. 600 has a total of 24 factors.
  2. 600 has only three prime factors which are 2,3 and 5.
  3. 600 can have one positive and one negative factor in pair.
  4. Pair Factors of 600 can have one prime and one composite number.

Solution

The product of one positive and one negative number is always negative. Hence 600 can never have one positive and another negative factor in pairs. So False statement is 600 can have one positive and one negative factor in pairs.

Images/mathematical drawings are created with GeoGebra.

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