Question

Find out whether $397$ is a prime number or not.

Answer 1

To find $397$ is a prime number or not, we need to find out how many prime factors are there of $397$. Because $397<400$, we check whether $397$ is divisible by any prime number less than 20.

So, the prime numbers less than $20$ are $2,3,5,7,11,13,17,19.$

Let us test the divisibility of $397$ by each of them.

Step 1- $397$ is not divisible by $2$because the digit in the ones place is odd.

Step 2- $397$ is not divisible by $3$ because $3+9+7=19$, but $19$ is not divisible by $3$.

Step 4- $397$ is not divisible by $5$ because the digit in the ones place is neither $5$ or $0$.

Step 5- $397$ is not divisible by $7$ because $397÷7$ gives quotient $56$ and remainder $5$.

Step 6- $397$ is not divisible by$11$because the difference of the sums of the digits at the alternate places $$\begin{gathered} 3+7=10 \\ 10-9=1 \end{gathered}$$ which is not divisible by $11$.

Step 7- Now, $397$ is not divisible by $13$ because $397÷13$ gives quotient $30$ and remainder $7$.

Step 8- $397$ is not divisible by $17$because $397÷17$ gives quotient $23$ and remainder $6$.

Step 9- $397$ is not divisible by $19$ because $397÷19$ gives quotient $20$ and remainder $17$.

Since, $397$ is not divisible by any prime number less than $20$, so $397$ is a prime number.