Question

# $\left( i \right)$ Express 49 as the sum of 7 odd numbers.$\left( {ii} \right)$ Express 121 as the sum of 11 odd numbers.

Hint: Find out all the odd numbers less than or equal to a given number, then find out 7 and 11 odd numbers whose sum is equal to 49 and 121 respectively.

As we know that the odd numbers are those who do not divide by 2.
$\left( i \right)$ Express 49 as the sum of 7 odd numbers.
So, first find out all the odd numbers less than or equal to the given number 49.
So, the odd numbers are
$1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47$
Now find out the seven odd prime numbers whose sum is 49 from the above odd numbers.
As we know 49 is the square of 7, therefore the first 7 odd numbers are the required numbers whose sum is 49.
Therefore seven odd prime numbers are $1,3,5,7,9,11,13$
$\Rightarrow 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49$
$\left( {ii} \right)$ Express 121 as the sum of 11 odd numbers.
So, first find out all the odd numbers less than or equal to the given number 121.
So, the odd numbers are
$1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63, \\ 65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111,113,115,117, \\ 119,121 \\$
As we know 121 is the square of 11, therefore the first 11 odd numbers are the required numbers whose sum is 121.
Therefore 11 odd numbers are $1,3,5,7,9,11,13,15,17,19,21$
$\Rightarrow 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 = 121$
So, this is the required answer.

Note: In such types of questions always remember the key concept that the odd numbers are those numbers which is not divisible by 2, then find out all the odd numbers less than or equal to the given number, then find out 7 and 11 odd numbers whose sum is equal to the given number respectively as above which is the required answer.
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