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Integer Function
Problem Statement
Find the smallest integer x for which



represents an integer.

(Source: Mathematics Teaching in the Middle School, Sep-Oct 1996)
.

Problem setup

This problem is asking me to find a number (the smallest possible) – here represented by x- that will make the fraction equal to an integer.

 

Plans to Solve/Investigate the Problem

Initially, I thought about the number 0; however, this is a sophomoric mistake of great proportions!!!  (Remember, I am full of drama!!)

 

Investigation/Exploration of the Problem

While using x = 0 will give me an integer result, 0 is not the SMALLEST value we can use.  Consider negative values.  Some of these will also give integer values.  Here is a chart for your review:

 

Value

Result

    1

6

0

12

-1

#DIV/0!

-2

-12

-3

-6

-4

-4

-5

-3

-6

-2.4

-7

-2

-8

-1.71429

-9

-1.5

-10

-1.33333

-11

-1.2

-12

-1.09091

-13

-1

-14

-0.92308

-15

-0.85714

-16

-0.8

-17

-0.75

-18

-0.70588

-19

-0.66667

-20

-0.63158

-21

-0.6

Notice that for several of the negative values, the result is indeed an integer!  When x = -13, the result is -1.  This is indeed the smallest value of x that will yield an integer result. 

 

This seems rather simple to me, so I would like to pose an additional question: What would cause the result to be zero? Let’s explore the world of calculus!

 

For an interactive lesson on this, click here.  Be sure to turn your sound on! 

Could we take the limit of this expression as x approaches infinity? 

 

    Ponder this  as the denominator grows more negative, the decimal result gets smaller!  As x approaches infinity (that sideways 8 over there represents infinity), the quotient gets closer and closer to 0.

Extensions of the Problem

Discuss possible extensions for the problem and explore/investigate at least one of the extensions you discussed.

Author & Contact
mhouston@rockdale.k12.ga.us

 

 

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