Download INVESTIGATION: DIVISIBILITY IN BASE n

Math 15
Spring 2012
INVESTIGATION: DIVISIBILITY IN BASE n
PROBLEM
Explore, describe, and explain the numerical patterns that arise in divisibility in number systems with
various bases. Specifically:
I
In base two, find a rule for how to tell whether a given number is even (that is, divisible by our
base-ten number 2) or odd. Do the same in base three, then in several other bases. Then write
a general rule about even and odd numbers to work for a number system of any base.
II Extend the above investigation by exploring divisibility in the following cases:
a) Base four: Find rules for divisibility by 2, 3, and 104.
b) Base six: Find rules for divisibility by 2, 3, 5, and 106.
c) Base seven: Find rules for divisibility by 2, 3, 6, and 107.
d) Base eight: Find rules for divisibility by 2, 4, 7, and 108.
III By comparing the results above together with divisibility rules in base ten, describe as many
generalizations as you can about divisibility. Look for general statements that are true within a
given base, and also for generalizations across two or more bases. In each case, do your best to
explain why your generalization holds.
Hint: Make sure you understand what is meant by “a divides b” [symbolized “a|b”] and divisibility rules within
the whole numbers in base ten. Read and reread Section 4.1 in the text!
PRESENTATION
Write a single group report which:
•
describes the problem and how your group went about solving it;
•
shows the processes and results of your calculations in easy-to-understand forms;
•
describes the patterns you found and the generalizations you drew from them; and
•
explains (if you can) why the generalizations are valid.
Your paper should be organized in the order (I, II, III) of the problem statement. Include sufficient
examples and explanation that the reader can follow your reasoning clearly. The report can take any
form or length you see fit, but it must address the problem adequately and be well-written and neat.
The text must be word-processed, though tables, calculations, and other graphics may be
handwritten.
ASSESSMENT
Your group’s report will be graded out of 90 points, based on both the mathematics (reasoning,
accuracy, completeness, depth) and the presentation (organization, explanation, writing quality,
appearance). Your group members will rate your participation for up to another 10 points.