A toymaker is making miniature 3-wheel and 4-wheel all-terrain vehicles (ATVs). He has a total of wheels and sets of handlebars. How many 4-wheel ATVs can he make?

Source: NCTM Math Teacher 2008

**SOLUTION**

If the toymaker makes 4-wheel ATVs, he uses wheels and sets of handlebars. The remaining wheel and handlebars are not enough to make any 3-wheel ATVs. If he makes 4-wheel, he uses wheels and handlebars. The remaining wheels and handlebars are enough to make 3-wheel ATVs leaving wheels and handlebars unused.

The following table summarizes the different combinations of ATVs

4-wheel 3-wheel wheels unused handlebars unused

Two possible solutions: 4-wheel and 3-wheel or 4-wheel and 3-wheel. Both solutions use all of wheels but the first one is better because it uses all of 17 sets of handlebars.

*Alternative solution*

The following figure shows the graph of the Diophantine equation

Two points and on the graph are lattice points.

**Answer**: