Pawnshop

The Pawnshop Problem

1. This problem comes from the seedy, smoke-filled office of pawnbroker Chuck Plastic, who insists on using cash in all his merchandise.

Today Mr. Plastic has sold $5000 worth of merchandise, purely from televisions and radios. He knows he has sold his televisions for $57 each and his radios for $32 each, but he doesn't remember how many of each he has sold. He's a bit slopy with his paperwork. Can you help Mr. Plastic by telling him how many of each he has sold?

A useful theorem applies here. In general an equation of the form ax + by = n (where x and y are the unknowns and are non-negative) is guaranteed to have at least one solution if :

1. a and b are relatively prime; and
2. n > ab - a - b

Generally the solution that you find to this equation has either x or y to be negative. By juggling the equation, you can also find solutions with both x and y positive.

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