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# Express the following number as the sum of odd primes 36.

Last updated date: 11th Sep 2024
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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.

Complete step-by-step solution:
We want to express 36 as a sum of two odd prime numbers.
Let us first write all prime numbers between 1 to 36.
The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19 23, 29, 31.
But according to the question, we only want the odd prime numbers.
So, the odd prime numbers are 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.
$\Rightarrow 36 - 3 = 33$
33 is not a prime number. So, this pair is rejected.
Next, let us try for 5.
$\Rightarrow 36 - 5 = 31$
31 is a prime number. So, the pair of required odd prime numbers are $\left( {5,31} \right)$.
Continuing similarly, we will get three more pairs of odd prime numbers which satisfy the required condition: $\left( {7,29} \right)$, $\left( {13,23} \right)$ and $\left( {17,19} \right)$.

Hence, we have got four pairs of odd prime numbers which satisfy the required condition:
$\therefore \left( {5,31} \right),\left( {7,29} \right),\left( {13,23} \right),\left( {17,19} \right)$

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.