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# Express 36 as a sum of two odd prime numbers.

Last updated date: 24th Mar 2023
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Hint: First find all the prime numbers from 1 to 36. Then remove the even prime numbers i.e. 2 from this list. Next subtract these odd prime numbers from 36 one by one. If the difference is a prime number, we have a pair of odd prime numbers which satisfy the required condition.

We want to express 36 as a sum of two odd prime numbers.
Let us first we write all prime numbers between 1 to 36.
Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19 23, 29, 31
But according to the question, we only want the odd prime numbers.
So, Odd prime numbers: 3, 5, 7, 11, 13, 17, 19 23, 29, 31
Now, we will subtract each of these odd prime numbers from 36 and check whether the difference is an odd prime number or not.
First, we will try for 3.
$36-3=33$ not a prime number. So this pair is rejected.
Next, let us try for 5
$36-5=31$. This is a prime number.
So we have a pair of the required odd prime numbers: $\left( 5,31 \right)$
$5+31=36$
Similarly, we can also find other pairs.
Let us continue the process.
Let us try on 7
$36-7=29$ which is a prime number
So we have another pair of required odd prime numbers: $\left( 7,29 \right)$$13+23=36$
$7+29=36$
Continuing in the similar fashion, we will get two more pairs odd prime numbers which satisfy the required condition: $\left( 13,23 \right)$ and $\left( 17,19 \right)$
So we have got four pairs of odd prime numbers which satisfy the required condition:
$\left( 5,31 \right);\left( 7,29 \right);\left( 13,23 \right);\left( 17,19 \right)$
Let us express 36 as a sum of these odd prime numbers:
$5+31=36$
$7+29=36$
$13+23=36$
$17+19=36$

Note: In the question, it is asked to find odd prime numbers. So do not forget to remove the even prime number: 2 from the list of all the prime numbers from 1 to 36 to get the list of all odd prime numbers from 1 to 36.