Question

What is the prime factorization of 108?

Answer 1

Hint: The given number is 108. The number given is even. Writing the prime factors means expressing the number as a product of prime numbers. We will start the prime numbers from 2. Then if 2 is not the factor of the given number we will move forward to get the next prime number that can divide the number. But the product should involve prime numbers only.

Complete step by step answer:
Now we are given a number 108. We know that this number is divisible by 2 since it is an even number. Thus the product becomes,
\[108 = 2 \times 54\]
Now 54 is again an even number that is divisible by 2.
\[108 = 2 \times 2 \times 27\]
Now the number 27 is not prime. So we will write it in the prime number form. But now it is not divisible by 2. So moving forward we will take the next prime number that is 3.

So 27 is absolutely divisible by 3. Thus the product becomes,
\[108 = 2 \times 2 \times 3 \times 3 \times 3\]
Since 27 is the cube of 3. Now all the numbers in the product form are prime. So we need not to further divide any number. This is the prime factorization form of the given number.
\[108 = 2 \times 2 \times 3 \times 3 \times 3\]

Therefore,the prime factorization of 108 is $ 2 \times 2 \times 3 \times 3 \times 3$.

Note: To solve this problem only the thing that is to be noted is that the numbers so chosen should be prime only. There is no bind that any prime number like 2 if once used cannot be used again. \[18 = 2 \times 3 \times 3\] like to express 18 as a prime factorization form. We took 3 twice but the product form includes all the prime numbers. This method of expressing a number into prime factor product form is used to find the LCM of numbers generally. In the problem above we can split or express 54 as a product of 3 and 18 also. Again 18 as products of 3 and 6. At last 6 as products of 3 and 2. But there would not be any effect or change on the final answer because the prime factorization is the same, only the way of factoring is changed.