Can 5 odd numbers be added to get 30?

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Answer 1 · 2015-09-10T20:50:34

No.

The sum of an odd number of odd numbers is odd.

Every odd number can be written as 2i+1# for an integrer, $i$ , so
For this question in particular, if we add:

$2i+1$
$2j+1$
$2k+1$
$2l+1$
$2m+1$

We get:

$2(i+j+k+l+m) +5$

$= 2((i+j+k+l+m+2) +1$

which is of the form $2n+1$ , so it is an odd number and cannot be equal to $30$ .

Answer 2 · 2015-09-10T21:06:18

In normal integer arithmetic - no.

In modular arithmetic - yes with any odd modulo > 30.

Normal arithmetic

Suppose your $5$ odd numbers are:

$a_1 = 2k_1 + 1$
$a_2 = 2k_2 + 1$
$a_3 = 2k_3 + 1$
$a_4 = 2k_4 + 1$
$a_5 = 2k_5 + 1$

where $k_1, k_2, k_3, k_4, k_5 in ZZ$

Then:

$a_1+a_2+a_3+a_4+a_5$

$= 2(k_1+k_2+k_3+k_4+k_5+2) + 1$

which is odd.

$30$ is even - not odd - so $a_1+a_2+a_3+a_4+a_5 != 30$

Arithmetic modulo $31$

Let $a_1 = a_2 = a_3 = a_4 = 13$ and $a_5 = 9$

Then $a_1+a_2+a_3+a_4+a_5 = 61 = 30$ modulo $31$

In fact in modular arithmetic modulo an odd base all numbers are both odd and even.