What five odd numbers add up to 100?

No 5 odd numbers add up to 100.

The sum of any two odd numbers is even.

The sum of any two even numbers is even.

So the sum of any 4 odd numbers must be even.

Take that and add an odd number and the sum must be odd.

Therefore, the sum of any 5 odd numbers must be odd.

100 is even.

In conclusion:

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