Prime Factorization: How to Find Factors of 72

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Factors of 72

When we multiply two numbers together, we get a product. The numbers that are multiplied are called factors. And the product is called the multiple. 

For example, 2 x 3 = 6

Here, 2 and 3 are factors, and 6 is multiple.

Factors are the numbers that divide another number without leaving a remainder. In the above case, 2 is a factor that divides 6 without leaving a remainder. Similarly, 3 is also a factor that divides 6 without leaving a remainder. So, 2 and 3 both are factors of 6.

To remember factors and multiples in an easy way, let us try to say, “Factors are Ninja. Multiples are Monsters”. Factors are ninja, which means they chop the numbers into smaller numbers. Multiples are monsters which means that they tend to grow bigger and bigger. Factors are finite, whereas multiples are infinite. Factors are always smaller than the number, or they can be equal to the given number. Multiples are always bigger than the number, or they could also be the same as the given number.

Let us now learn how to find positive factors and multiples of a number.

Finding factors of a prime number

There are always only 2 factors of a prime number. The first factor is 1, and the other factor is the number itself. A prime number is a number that is divisible only by 1 or the number itself, and so, it has only 2 factors.

Finding factors of a composite number

There are always more than 2 factors of any given composite number. There are several ways to find factors of a number. Let us take a number 72 and try to find the factors of it using different methods.

Factors of 72

  • Pair Factorization method

In this method, we start from 1 and then see which number should be multiplied by it to get the result 72. So we write as,

            1 x 72 = 72

Then, we take the number 2 and check if 72 is divisible by 2. If yes, then find out which number should be multiplied to 2 to get 72. So we get,

            2 x 36 = 72

Then we take the next number, 3 and follow the same process. So, we get,

            3 x 24 = 72

Then we take the next number, 4 and follow the same process. So, we get,

            4 x 18 = 72

Then we take the next number, 5, but we find that 72 is not exactly divisible by 5. So we skip 5 and take the next number 6. 72 is divisible by 6. So, we get,

            6 x 12 = 72

Then we take 7, but 72 is not divisible by 7. Then, take 8. 72 is divisible by 8. So, we get,

            8 x 9 = 72

Then the next number is 9. But we can see above that 9 is already taken while writing 8×9. So, we will stop here. We do not need to take any number further.

Hence, the pair factors of 72 are: 

1 and 72, 2 and 36, 3 and 24, 4 and 18, 6 and 12, 8 and 9.

We can also say that the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72

  • Prime factorization method

In the prime factorization method, we only find the factors which are prime.

For example, factors of 72:

Here, first, we write 72 and then divide it by the 1st possible prime number. The 1st prime number is 2, and since 72 is divisible by 2, we start with 2. 72/2 = 36. So, we write 36 below 72 and then again take a prime number which can divide 36. We choose 2. 36/2 = 18. So we write 18 below 36. 

Then again, divide 18 by any prime number. We choose 2. 18/2 = 9. So we write 9 below 18. Then again, divide 9 by any prime number. Here 2 is not possible, and so we choose 3. 9/3 = 3. So we write 3 below 9. Now we got 3 at the end, which is a prime number itself. So, we can stop factoring here.

So, the prime factors of 72 can be written as 2 x 2 x 2 x 2 x 3 x 3.

The prime factorisation is most widely used as we need to find the HCF (Highest Common factor) of 2 or more numbers. This can be only done by finding their prime factors first. Then we look at the factors which are common and then multiply them. 

For example, to find HCF of 24 and 72

We first need to find the prime factors of 24. The prime factors of 24 can be written as 2 x 2 x 2 x 3

Then we need to find the prime factors of 72. The prime factors of 72 can be written as 2 x 2 x 2 x 2 x 3 x 3.

In the prime factors of both the numbers above, some factors are common. Let us write them separately. They are 2,2,2,3. Now we should multiply them to get the HCF.

Therefore, HCF = 2x2x2x3 = 24

We can also note that the HCF of 2 or more numbers is always equal to the smallest number (in our example, it was 24) or smaller than the smallest given number. 

  • Factor tree method

In the factor tree method, we branch out the number into pair factors. For example, Factors of 72 

This method is a combination of pair factorisation and prime factorisation. Here, we look for a prime factor and then write its pair at the same level. Like, 2 is a prime factor of 72, and 36 is its pair factor. We write them in circles at the same level. Then we keep on factoring all the numbers till we get both prime factors at the end. 

Finding multiples of a number

Multiple of a number could be the same number or bigger than that. We get multiples when we multiply the given number by any number. The resultant product is called multiple. 

For example – 

Multiples of 2 are 2, 4, 6, 8, 10, 12, etc

Multiples of 3 are 3, 6, 9, 12, 15, etc

We can see that the multiples are growing. We can also see that multiples of a number can be infinite. In simple words, multiples are the numbers that we get on the right-hand side (RHS) while writing the table of a number. 

Multiples are used to find the Lowest Common Multiple (LCM) of 2 or more numbers. For example, in the above case where we found multiples of 2 and 3, 6 is a common multiple. There could be more common multiples (e.g., 12 in the above case), but still, 6 is the lowest common multiple. Hence, the LCM of 2 and 3 is 6.

Conclusion

We have so far discussed factors and multiples in detail. We got to understand them in detail. We also learned how to find them and where they are used. I hope you practice more sums to understand them thoroughly.

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