Clearing your basics is the most important part of better learning. To do that, you need to understand basic terminology. Prime numbers are natural numbers that are only divisible by 1 or by themselves. It means that these numbers are not equal to m x n where m and n are any integers (except 1). The prime numbers begin from 2 because 1 is neither prime nor composite by nature. However, Euclid, the famous classical era mathematician, recorded proof that there is no limit to the “biggest” prime number.
Therefore, before we jump into the question at hand to look at Pair Factors of 72, it is essential to look at prime numbers.
Finding smaller prime numbers
There are some basic steps you can follow easily To find small prime numbers.
Step 1: Write down a list of whole numbers from 2 to n, where n is the upper limit of the numbers you want to test.
Step 2: Next, starting from 2, erase every number that is a multiple of 2.
Step 3: For step 3, you can repeat the same step with any following remaining number and erase all numbers that are multiples of it.
Step 4: The remaining numbers that are not divisible by any numbers smaller than them in the list are your prime numbers.
What is Prime Factorisation?
The prime factorisation process is a multi-step process to break down any number to its fundamental prime number. This implies that every number can be shown as a multiple of various prime numbers.
Essentially, dividing and redividing a number with the help of smaller prime numbers that the number is a multiple of until a point where all of its factors are prime numbers is how you perform prime factorisation.
For instance, the number 35 is the multiple of two prime numbers: 5 and 7. You can calculate all the prime factors of any given number with the process of prime factorisation.
Performing prime factorisation:
You can follow these steps to perform prime factorisation on any number. Let’s say the number in question is represented by n.
Step 1: Find the smallest prime number by which the number n is divisible. For instance, if the number is 39 (n=39), it is not completely divisible by 2, but it is divisible by 3. Thus, 3 is the first prime number that can completely divide 39.
Step 2: This can be easily represented in the form of a table, like below:
| 3 | 39 |
Step 3: Next step is to divide n by the prime number and put the factor in the table’s next line, like so:
| 3 | 39 |
| 13 |
Step 4: The next step is to repeat steps 2 and 3 until the remaining factor is also a prime number.
Step 5: once the remaining factor is also a prime number, you can divide it by itself and leave the remaining factor as 1. Thus, giving you all the prime factors of the number n. In the example used, it can be represented like so:
Learning more examples
Here are a few more numbers solved as a prime factorisation table:
Solve 28 for its prime factors.
Solution:
| 2 | 28 |
| 2 | 14 |
| 7 | 7 |
| 1 |
The prime factors of 28 are 2, 2 and 7.
Solve 36 for its prime factors.
Solution:
| 2 | 36 |
| 2 | 18 |
| 3 | 9 |
| 3 | 3 |
| 1 |
The prime factors of 36 are 2, 2, 3 and 3.
Solve 52 for its prime factors.
Solution:
| 2 | 52 |
| 2 | 26 |
| 13 | 13 |
| 1 |
The prime factors of 52 are 2, 2 and 13.
Solve 68 for its prime factors.
Solution:
| 2 | 68 |
| 2 | 34 |
| 17 | 17 |
| 1 |
The prime factors of 68 are 2, 2 and 17.
Solving prime factorisation for finding the factors of 72
Here is how you solve for the factors of 72
Step 1: Find out what is the smallest prime number that can divide the number 72. Since it is divisible by 2, that is the smallest prime number that can divide 72.
Step 2: Enter the remaining factor value into a table. You can do this like so:
| 2 | 72 |
| 36 |
Step 3: Since the remaining factor is 36, we know it is not a prime number. Therefore, you should divide it further with the smallest prime number. 36 is divisible by 2, and thus, that is the smallest prime number that can completely divide 36.
Step 4: Enter the remaining factor value into a table. You can do it like so:
| 2 | 72 |
| 2 | 36 |
| 18 |
Step 5: Since the remaining factor is 18, we know it is not a prime number. Therefore, you should divide it further with the smallest prime number. 18 is divisible by 2, and thus, that is the smallest prime number that can completely divide 18.
Step 6: Enter the remaining factor value into a table. You can do it like so:
| 2 | 72 |
| 2 | 36 |
| 2 | 18 |
| 9 |
Step 7: Since the remaining factor is 9, we know it is not a prime number. Therefore, you should divide it further with the smallest prime number. 9 is not divisible by 2; however, it is divisible by 3. Thus, 3 is the smallest prime number that can completely divide 9.
Step 8: Enter the remaining factor value into a table. You can do it like so:
| 2 | 72 |
| 2 | 36 |
| 2 | 18 |
| 3 | 9 |
| 3 |
Step 9: Since the remaining factor is 3, we know that it is a prime number. Therefore, you can not further divide it with any other number but itself. Thus, 3 is the smallest number that can completely divide 3.
Step 10: Enter the remaining factor value into a table. You can do it like so:
| 2 | 72 |
| 2 | 36 |
| 2 | 18 |
| 3 | 9 |
| 3 | 3 |
| 1 |
Step 11: Now that the remaining factor value is 1, we know that the prime factors of 72 are 2, 2, 2, 3 and 3. You can also represent these factors as an equation in the form of multiplication where,
2 x 2 x 2 x 3 x 3 = 72.
Conclusion:
From this exercise, we can conclude that the prime factors of 72 by the prime factorisation method are 2, 2, 2, 3, and 3. Similarly, you can calculate the prime factors of any composite number by the prime factorisation method.
