A point is called integral if both and are integers. How many points on the graph of are integral points?

Source: NCTM Mathematics Teacher, November 2006

**SOLUTION**

Divide by

,

Divide by

Divide by

, no solution

, duplicate solution

Divide by

, no solution

Divide by

, no solution

, duplicate solution

, duplicate solution

Divide by

, no solution

, no solution

, no solution

Divide by

, no solution

, no solution

, no solution

, duplicate solution

We found five solutions . We need an algebraic way to find all the solutions.

For to be an integer must divide . It’s positive divisors yield the five previous solutions. The negative divisors yield four more solutions, namely . One solution is discarded because and . Nine points on the graph are integral points.

**Answer**:

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Retired high school math teacher.