I can understand you mysterious user80586.
One example is Latin squares and its application in design experiment. You can search in the web for more information about it.
Also I give a suit example that your audiences love it. The number of divisors of a small number is very important. it is a pure number theoretic problem. But, suppose a company want to packing its production like as chocolate or biscuit. They prefer to use number 4, 6 or 12. Why? for better sharing the chocolates or biscuits in a package among some friends.
You can see the very interesting book with name:
"I' explosion des Mathematiques" that is in French language.