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Module 1 Homework – 3303
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This is a 47 point assignment.
Homework rules:
Front side only. Keep the questions and your answers in order.
Send it in ONE pdf file to my email inbox by the deadline. ONE FILE, please.
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Question 1:
6 points (1, 1, 1, 3)
Multiplication and polynomials
In Chapter 1 p. 2, the authors assert
“Thus our number system is a shorthand for the algebra of polynomials in one variable
with the variable replaced by the number 10.”
A.
What is a polynomial?
What is the “algebra of polynomials”?
B.
How can you represent the following numbers as polynomials in x?
17
100
1,527
C.
How do you multiply 490 by 102 using the representative polynomials?
Show every step.
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D.
In a short*, attached essay, expand upon the statement:
“Thus our number system is a shorthand for the algebra of polynomials in one variable
with the variable replaced by the number 10.”
Things to consider:
Can you represent 7½ as a polynomial? What about 57? or 3 ?
5x 3  3x  2 : what number does this stand for? What are some alternate representations
of this number?
Does every polynomial in the world stand for a number – or – is it that every number can
be represented by a polynomial? Or something in-between? What might be a better
description of “number system”? Do the authors mean all real numbers?
Would you like to re-teach multiplication this way to reinforce polynomial multiplication
to, say, juniors in high school? or seniors? Do you think it might be effective?
*short: one page front side only, typed in 12 point type with ½“ margins.
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Question 2:
2 points
In Chapter 1, p. 8, the authors discuss Roman numerals.
A.
Discuss why they wrote the sentence:
“First one changes all numbers into the form in which no subtractions are indicated.”
Why do they say that and how do you do this? Illustrate with MCMXCIX = 1,999
B.
Show how you add MCMXCIX and CD.
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Question 3:
6 points
On page 13, the authors discuss a way to tell if a number, abcd, is divisible by 3, 9, or 11.
A.
Write a brief discussion below of why separating the number as they did on page
13 provides a test for divisibility by 11. Shoot for understanding by a 12 year old.
B.
Use their method to discuss how you would do something similar to see if a
number, abcde, is divisible by 4. Be very clear.
hint:
There’s a nice discussion of divisibility at http://www.jimloy.com/number/divis.htm and
a pdf on my website from there. It’s slightly different from the authors’ presentation but
is still a nice summary.
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Question 4:
6 points (2, 2, 2)
On pages 13 – 17, the authors discuss “induction”. I have put a chapter from David
Cohen’s Precalculus on my website for you to read.
Please work problems 4,
9,
and 20.
5  9  ...  (4n  1)  n(2n  3)
(3n 1  3)
2
3
n  2n is divisible by 3
3  32  ...  3n 
Do it quite formally, showing each step in the induction process use one page per
problem. Attach the extra sheets to this one in order.
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Triangular Numbers – put the illustrations on attached sheets,
appropriately numbered.
13 points (2, 2, 2, 3, 2, 1, 1)
Question 5:
A.
Show two examples each of the following facts* about triangular numbers
i.*
A number is triangular if and only if it is of the form:
n(n  1)
for some n  1 . [proof: Pythagoras, circa 440 BC]
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ii.*
The integer n is a triangular number if and only if 8n + 1 is a perfect
square.
[proof: Plutarch, circa 100 AD]
iii.
The sum of any two consecutive triangular numbers is a perfect square.
[proof: Nicomachus, circa 100 AD]
iv.
If n is a triangular number, then so are 9n + 1, 25n + 3, and 49n+6.
[proof: Euler, 1775]
* “if and only if” is a contraction of two conditionals. Please illustrate each conditional
statement with different numbers for a total of 2 examples for each for i and ii.
B.
Find the following and list them:
i.
Two triangular numbers whose sum and difference are also triangular
numbers.
ii.
Three successive triangular numbers whose product is a perfect square.
iii.
Three successive triangular numbers whose sum is a perfect square.
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Question 6:
Abundant and deficient numbers, p. 23
3 points
What is a “proper divisor”?
Can you find two definitions of proper divisor on the internet? Which are we using in
this class?
Show that 12 is an abundant number. Be sure to show it in detail.
Give an illustration of your choosing – different from everybody else’s – of another
abundant number.
Show that 15 is a deficient number – also in detail.
Give an example – different from everybody else’s – of another deficient number.
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Question 7:
Primes
4 points
Attach the answers, suitably labeled on addition sheets after this one.
A.
You can find out if a number, x, is prime by testing all primes  x as possible
divisors. If you find none that are divisors, then x is prime. Why does it work to
use the square root of x and not just x as the boundary on primes that you must
test?
B.
Illustrate the fact that any composite three-digit number must have a prime factor
less than or equal to 31. Use a number different from the people you help or who
help you, please. I want to see LOTS of different illustrations – a different
example on each homework turned in.
C.
A twin prime pair is a set of primes that are of the form n and n+2 where both
numbers are prime. Mathematicians have discovered over 150,000 twin primes.
Please find 3 pairs of twin primes. (hint: For primes greater than 3, the sum of
twin primes is divisible by 12).
D.
In 1950 it was proved that any integer greater than 9 can be written as a sum
of distinct odd primes. Show that this is true for 25, 66, and 81. [ “distinct”
means different from one another, not repeated].
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Question 8
4 points
Repunits
A.
Give the definition of a repunit.
Show a number that is clearly a repunit and one that is not.
B.
Give a mathematical expression that defines the number of natural numbers
between these two repunits: Rn and Rn+1.
C.
Find a repunit with a prime index that is composite
(different from the one I did in lecture).
D.
Show a repunit with a composite index and show the factors.
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Question 9
3 points
Geometric progressions.
Give a definition of a geometric progression, along with two different examples.
Cite your source.
Give the formula for the sum of the first n numbers in a geometric progression. Show
this by the formula and actual addition on your examples for n > 5.
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Question 10
3 points
Summarize the Euclidean algorithm in a brief essay in such a way that a 5th grader could
get it. Use an illustration different from the one in the book and the one below. Type the
essay on an attached sheet of paper. Treat it like a mini-lecture.
Demonstrate the Euclidean algorithm for finding the gcd of (105, 175) immediately
below:
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Question 11
3 points
Efficiency and signage.
Here’s a symbol with boxes where all the possible places to put instructions are.
Note: front, back, top, bottom, upper left and right, lower left and right…eight places to
hang numbers or symbols to give information or instructions to do something.
No adding or subtracting allowed…only adjacency, multiply or divide.
Find math or science examples that use each place, singly or in combination.
Explain what each one means in the example you’ve chosen.
Symbol
I’ll get you started:
52 uses the box in front for multiplication by 1 and the
upper left box for squaring.
This is an alternative representation for the integer 25.
and
Cn the combination of k things taken n at a time. You
n!
calculate this with the formula:
. Alternate
k!(n  k)!
notation is Cnk . 6 C3  20 .
k
See how many of these you can find. Put the symbol and the explanation on an attached
sheet or sheets.
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