Question

Show that every odd prime can be put either in the form \[4k+1\] or \[4k+3\] where, \['k'\] is a positive integer.

Answer 1

Hint: We solve this problem by assuming the odd prime number as a variable then we divide it by 4 to get all the possible formations of odd prime when divided by 4 that is we take all the possibilities of the remainder when a number is divided by 4.
We use the condition that the remainder of a division will always be greater than or equal to 0 and less than the divisor.
Then we check whether all the possibilities satisfy the definition of a prime number.
A number is said to be the prime number if it has only two factors that are 1 and itself.
By using this definition we find the general form of an odd prime number.

Complete step by step answer:
Let us assume that the odd prime number as \['n'\]
Now let us divide the number \['n'\] by 4 such that we get a remainder \['r'\] so that we get
\[\Rightarrow n=4k+r\]
Here, we can say that the number \['k'\] is some positive integer.
We know that the remainder of a division will always greater than or equal to 0 and less than the divisor.
By using this condition we get the domain of \['r'\] as
\[\Rightarrow 0\le r<4\]
Here we can see that \['r'\] is a integer. So, we can have the possible values of \['r'\] are 0, 1, 2, 3.
Now, let us take the first value that is \[r=0\] then we get the odd prime number as
\[\Rightarrow n=4k\]
Here we can see that the number \['n'\] has more than 2 factors that is 4 and \['k'\]
But, we know that a number is said to be the prime number if it has only two factors that are 1 and itself.
So, from the definition we can say that \[n=4k\] is not possible.
Now, let us take the second value that is \[r=1\] then we get the odd prime number as
\[\Rightarrow n=4k+1\]
Here, we can say that this form satisfy the definition of prime number because there are only two factors for \['n'\]
So, we can say that representing the odd prime as \[4k+1\] is acceptable.
Now let us take the third value that is \[r=2\] then we get the odd prime number as
\[\begin{align}
  & \Rightarrow n=4k+2 \\
 & \Rightarrow n=2\times \left( 2k+1 \right) \\
\end{align}\]
Here we can see that the number \['n'\] has more than 2 factors that is 2 and \['2k+1'\]
So, from the definition of a prime number we can say that \[n=4k+2\] is not possible.
Now, let us take the second value that is \[r=3\] then we get the odd prime number as
\[\Rightarrow n=4k+3\]
Here, we can say that this form satisfy the definition of prime number because there are only two factors for \['n'\]
So, we can say that representing the odd prime as \[4k+3\] is acceptable.
Therefore, we can conclude that the odd prime number can be represented in the form \[4k+1\] or \[4k+3\] where, \['k'\] is a positive integer.
Hence the required result has been proved.

Note: We have the other explanation for this problem.
We know that all the prime numbers are odd except 2.
We are asked to represent the odd prime number.
So, by removing one and only even prime number 2 we get all the remaining prime numbers are odd.
We know that the odd number is represented in the form \[4k+1\] or \[4k+3\] where, \['k'\] is a positive integer.
Since we know that all primes are odd numbers we can say that the representation of odd numbers will also satisfy the representation of odd prime numbers.
Therefore, we can conclude that the odd prime number can be represented in the form \[4k+1\] or \[4k+3\] where, \['k'\] is a positive integer.
Hence the required result has been proved.