Explain whether $3\times 12\times 101+4$ is a prime number or composite number.

Question

Vedantu Content Team · Accepted Answer

Hint: We have to find whether $3\times 12\times 101+4$ is a prime number or composite number. For that we know, every prime number has only two factors. Try to recollect the definition of prime number, composite number. Then, find the given number and analyse it, we will get the answer.

Complete step-by-step answer:
A prime number is a number which has exactly two factors, which are 1 and the number itself. We don’t consider 1 as a prime number. Also, we know that a composite number has more than two factors, apart from 1 and the number itself. We don’t consider the number 1 as a composite number.
For example, if we consider a number 6, then we know that it has factors as 1, 6, 2, 3, which are total four in number. So, we can say that 6 is a composite number. Now, let us consider the number 5, then we know that it has only two factors as 1 and 5. We can clearly say that it is a prime number as it has only 1 and number itself as the factors.
Now we are given a number $3\times 12\times 101+4$ in the question. So, first, it is necessary to know how many factors this number has. We can find this out by taking out common factors and factoring the given number.
We can take 4 outside from both these terms and we will get $4\left[ 3\times 3\times 101+1 \right]$ . Now, we can multiply the terms inside the bracket and we will get $4\left[ 909+1 \right]$ . Adding both the terms, we get $4\times 910$ . Now, we know that 4 has factors 1, 4, 2, 2 and we can see that 910 also has more than 2 factors as it is an even number. So, in short, we can say that the given number $3\times 12\times 101+4$ has more than 2 factors.
Therefore, $3\times 12\times 101+4$ is a composite number.

Note: Read the question and understand what is asked. Your concept regarding the prime numbers and the composite numbers should be clear. The given number might appear very complex and huge, but it is necessary to simplify it and then check the number of factors it has. Do not make silly mistakes while factorising.