Warmup #4 for Tue 1-9-07

If n is a positive integer from 1 to 144 inclusive, how many fractions of the form n/144 are in lowest terms? For example, 1/144 is in lowest terms and 2/144 is not.

Grades: 5 and up
Answer: 48

Comments and Possible Solution: [Change the denominator to 12, 24 or 36 to make it a quicker warm up.] This is a fairly well-known type of problem that is accessible to students at many levels. It often appears as a math contest problem, but it can be the basis for a class activity. A 5th grader can, with a partner, list every fraction and cross out the reducible ones. Most students quickly recognize that they can eliminate all even numerators. From the remaining 72 numerators, the odd multiples of 3 (3,9,15,...) can be eliminated. There are 144/3 = 48 multiples of 3 in all, half of which would be odd. Eliminating these 24 numbers leaves 48 values.

Additional comments: There are many approaches here. The idea of counting the reducible fractions and subtracting is a powerful strategy that does not occur to everyone at first. In the upper middle grades and high school, a problem like this can be used to develop number theory concepts, such as the concept of relatively prime and Euler's phi function. Number- theoretic ideas need more attention in middle school IMHO. See if you can find this in the new NCTM Curriculum Focal Points document.