Question

Express each of the numbers as the sum of three odd primes: $\left. 1 \right)21$ $\left. 2 \right)31$ $\left. 3 \right)53$ $\left. 4 \right)61$

Answer 1

Hint:
In this question first of all we will write all the odd numbers which are less than $50$ and then we will find the triplets of odd numbers which in addition gives the numbers which are asked in the question. Similarly we will do for all the numbers asked in the question.

Complete step by step solution:
The odd numbers which are smaller than $50$are $1,\,3,\,5,\,7,\,9,\,11,\,13,\,15,\,17,\,19,\,21,\,23,\,25,\,27,\,29,\,31,\,33,\,35,\,37,\,39,\,41,\,43,\,45,\,47,\,49\,$. Now, we will find three odd numbers which in addition gives the number asked in the question.
First we will find three odd numbers for $21$ . If we add $7,\,3$ and $11$ we will get $21$ . Hence, $21$ can be expressed $7 + 3 + 11 = 21$.
We will find three odd numbers for $31$. If we add $3,\,11$ and $17$ then we will get $31$. Hence, $31$ can be expressed as $3 + 11 + 17 = 31$.
Now, we will find three odd numbers for $53$. If we add $13,\,17$ and $23$ then we will get $53$. Hence, $53$ can be expressed as $13 + 17 + 23 = 53$ .
Finally, we will find three odd numbers for $61$. If we add $13,\,19$ and $29$ then we will get $61$.

Hence, $61$ can be expressed as $13 + 19 + 29 = 61$.

Additional information:
Odd numbers are those which on division with two leaves a remainder.

Note:
There can be more than one way to form a given number by the addition of three odd numbers. Example when
$9,\,1$ and $11$ are added it will give $21$. So just be careful about these things. We should be able to find the three odd numbers which can form a given number when these three odd numbers are added.