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How many two-digit prime numbers are there having the digit $ 3 $in their unit place?
(A)$ 10 $ (B)$ 8 $ (C)$ 6 $ (D)$ 5 $

Answer Verified Verified
Hint- Prime numbers are numbers that have only $2$factors, $1$ and themselves. For example, some prime numbers are$2$, $3$, $5$and $7$. To check whether a particular number is prime or not we test divisibility up to square root of the number, if a number is not prime we will find one of the factors of the number.


Complete step by step solution-
The number of two-digit numbers from$ 10 $ to $ 99 $ = $ 100 $
The number of two digit numbers having digit $ 3 $ in their unit place \[(13,23,33,43,53,63,73,83,93)\]= \[10\]
We have to find the prime numbers from the above \[10\] numbers; therefore we need to check these numbers one by one that they are prime numbers or not. To check whether a number is a prime number we have to check that the number is divisible by the prime number less than the square root of itself or not.
1)$ 13 $is not divisible by any of the prime numbers less than $ \sqrt {13} $.
2)$ 23 $ is not divisible by any of the prime numbers less than $ \sqrt {23} $.
3)$ 33 $is clearly divisible by $ 3 $(which is less than $ \sqrt {33} $), we can check this by applying the divisibility rule of $3$.
4)$ 43 $is not divisible by any of the prime numbers less than $ \sqrt {43} $.
5)$ 53 $ is not divisible by any of the prime numbers less than $ \sqrt {53} $.
6)$ 63 $ is clearly divisible by $ 3 $( which is less than $ \sqrt {63} $), we can check this by applying the divisibility rule of $ 3 $.
7)$ 73 $ is not divisible by any of the prime numbers less than $ \sqrt {73} $.
8)$ 83 $is not divisible by any of the prime numbers less than $ \sqrt {83} $.
9)$ 93 $is clearly divisible by $ 3 $( which is less than $ \sqrt {93} $), we can check this by applying the divisibility rule of $3 $.
So, there are a total of 6 prime numbers in the given range

So,option C is the right answer

Note: There is no generalized formula to check whether a number prime or not. First you need to shortlist the number by given condition (for example in our question we have shortlisted the numbers by the condition that it should be a two-digit number having the last digit ‘$ 3 $’) and check one by one that it is prime number or not.

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