What five odd integers have a sum of $30$?

Solution 1:

As straightforward mathematics there is no answer.

As anyone who has ever placed hymn numbers in a hymn board will know, it is possible to turn $9$ upside down to get $6$, and if this is allowed by the wording you can get a sum of $30$.

Likewise if it is odd numbers which are chosen, but the digits rather than the numbers which are added, the set $3,5,7,9,15$ gives $3+5+7+9+1+5=30$ - again this depends on precisely how the question is worded.

Solution 2:

By definition, odd integers are of the form $2n+1$ where $n\in \mathbb{Z}$. Since we want the sum of $5$ odd integers to be equal to $30$ this would imply that for $a,b,c,d,e \in \mathbb{Z}$ $$(2a+1)+(2b+1)+(2c+1)+(2d+1)+(2e+1)=2(a+b+c+d+e+2)+1=30$$ which is impossible since $(a+b+c+d+e+2)\in \mathbb{Z}$ and $2(a+b+c+d+e+2)+1$ is an odd integer by definition.