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# If a is a Natural number and $1\le a\le 20$, then find the number of prime numbers of the form 5a + 1. \begin{align} & a)4 \\ & b)5 \\ & c)6 \\ & d)7 \\ \end{align}

Last updated date: 12th Aug 2024
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Hint: Now to solve the given problem we will first check the value of 5a + 1 for all odd values of a. Now we know that even numbers which are greater than 2 are not prime. Now we will check the values when we substitute even values of a. hence we will get a sequence of numbers. Now we will count the prime numbers in the sequence.

Now we know that $1\le a\le 20$
Now we want $5a+1$ to be a prime number.
First let us understand prime numbers. Prime numbers are numbers which cannot be divided by any number other than one and the number itself.
For example take the number 5. 5 is divisible by the number itself and 1. There is no other number which is divisible by 5.
Now similarly if we take number 21 then the number 21 is divisible by 7 and 3 hence 21 is not a prime number.
Now we want 5a + 1 to be a prime number.
Now let us substitute a = 1 in the equation. Then we get 5a + 1 = 6. Now 6 is not a prime number.
Now if we notice for all odd values of a, the number 5a will be odd and hence 5a + 1 will be even.
Now we know that all even numbers are divisible by 2. Hence the number will never be prime in this case.
Hence let us just check the even values of a which are a = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
Now for a = 2, 5a + 1 =11
For a = 4, 5a + 1 = 21
For a = 6, 5a + 1 = 31.
Hence now if we notice the sequence we will get 11, 21, 31, 41, 51, 61, 71, 81, 91, 101.
Now we know that the only prime numbers among these are 11, 31, 41, 61, 71, 101.
Hence we will get 6 prime numbers.
Hence option c is the correct option.

Note: Now note that 2 is the only even prime number, rest all even numbers are non-prime. But all odd numbers need not be prime. For example 21 is odd but not a prime number. Also note that addition of prime numbers need not be prime. Hence when we have a + b to be a prime number this does not mean a and b are both prime numbers.