THE OPEN-ENDED CONTEST PROBLEM AND SOLUTIONS

As promised, here is the open-ended, rubric-based, holistically scored, performance-assessed, student-constructed first problem from MathNotation's Third Contest:

1. A primitive Pythagorean triple is defined as an ordered triple of positive integers (a,b,c) in which a² + b² = c² and the greatest common factor (divisor) of a, b and c is 1. If (a,b,c) form such a triple, explain why c cannot be an even integer.

Comments
(a) The content here is number theory. Is some of this covered in your district's middle school curriculum or beyond? More importantly, at what point do students begin to formulate and write valid mathematical arguments?

(b) The immediate reaction of most students was that this seemed like a fairly simple problem. However, only a couple of teams scored any points. Perhaps the challenge here was the construction of a deductive argument, although as you will see below, there is one challenging part.

(c) There were two successful approaches used by the teams. Both involved indirect reasoning. Do your students begin to do these in middle school or are "proofs" first introduced in geometry?

(d) I allowed students to assume without proof the following:

(i) The general rules of parity of the sum of two integers
(ii) The square of a positive integer has the same parity as the integer

(e) Interestingly, none of the teams considered an algebraic approach to the one challenging case, i.e., demonstrating that the sum of the squares of two odd integers is not divisible by 4.

If a and b are odd, they can be represented as
a = 2m+1 and b = 2n+1, where m and n are integers.
Then a² + b² = (2m+1)² + (2n+1)² =
(4m² + 4m + 1) + (4n² + 4n + 1) =
4(m² + n²) + 4(m + n) + 2, which leaves a remainder of 2 when divided by 4.
BUT, if c is even, say c = 2k, then c² = 4k², which is divisible by 4.

(f) The two best solutions came from our first and second place teams, Chiles HS in FL and Hanover Park Middle School in CA. Both used the ideas of congruence modulo 4.

Here is the indirect method used by Chiles:

Let's assume that c can be an even integer. We'll prove by contradiction. An even integer can be summed in two ways:
1. with two even integers or
2. two odd integers
If it is the latter case, then looking at the residuals of modulo 4, the two odd integers summed will be equal to 2, but this is not the case as 2 is not a modulo of 4 residue. If it is the former case, then it does not satisfy the problem as then a, b, and c have common factor of 2. Therefore c must be an odd integer. Q.E.D.


Here is the indirect method used by Hanover Park:

Suppose, for the sake of contradiction, that there is a PPT (primitive Pythagorean Triple) s.t. c is even. Then c² ≡ 0 (mod 4).

We break this into cases based on the parity of a,b.

Case I: Both a and b are even; gcd(a,b,c) ≥ 2 because a,b,c are even, a contradiction.

Case 2: One of a and b is even. Then, a² + b² ≡ 0 + 1 ≡ 1
not ≡ 0 (mod 4), a contradiction.

Case 3: Both of a, b are odd. Then a² + b² ≡ 1 + 1 ≡ 2
not ≡ 0 (mod 4), a contradiction.
We have covered all cases for a, b with no valid cases. Thus, in a PPT, c cannot be even.

Both of these arguments represent a more sophisticated understanding of mathematics and the methods of proof. Clearly, these students are quite advanced and exceptional, however, I feel many middle school teachers begin early on to encourage their students to explain their thought processes both orally and in writing. Am I right? I would like to hear your thoughts on this...

Using Number Theory To Promote Logic and Writing in Middle Schoool and Beyond

The following examples also provide practice for open-ended questions and a view of the Explain or Show type questions on our next Online Math Contest to be held in 5 weeks (see info below). Since formal proof is not the goal here, students are encouraged to write a logical chain of reasoning in which they can use/assume basic knowledge about odd and even integers. Further, these questions strongly suggest the strategies consider a simpler case first and patterning.

Another benefit of these types of questions is to review important terminology and to help students improve reading comprehension, a major obstacle for many youngsters in math class (and everywhere else!). Some middle schoolers and high schoolers will have difficulty making sense out of what the question is asking because of both the wording and the information load in the problem. We need to help them group key phrases together and, yes, I guess that means we are also reading teachers!

Example 1
Is the sum of the squares of the first 2009 positive integer multiples of three odd or even? Explain your reasoning.

Example 2
Is the sum of the squares of the first 2010 positive integer multiples of three odd or even? Explain your reasoning.

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REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

Updates (Pls Read!!)
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.

Taking Middle Schoolers Beyond Procedures To The Next Level...

Typical Classroom Scenario?
We're introducing the idea of least common multiple of two positive integers and after defining the terminology and illustrating several examples most students are catching on to some procedural method of which there are many:
Listing common multiples of each
Prime Factorization
The "upside down division method" you saw at a conference...

Yes, we are all very good at demonstrating step by step procedures and having students practice repetitively until they catch on and can reproduce this with some speed and accuracy. We feel this is a worthwhile skill (they'll need it for common denominators, clearing denominators in rational equations, useful for solving certain types of word problems, etc), it's in the curriculum and the standards, it will be tested in various places and the lesson plays out. Some students pick up the method(s) quickly, while others struggle, particularly those who haven't learned their basic facts.

BUT how can we raise the bar to stretch their minds? Can the above scenario be restructured to enable students to gain a deeper understanding of the concepts of lcm and gcf? Perhaps we can start the class off with a more open-ended type of question and ask them to work in small groups to solve it. Perhaps, we can ask a different type of question after teaching some standard procedure. A nonroutine, higher-order question that is not in the text...

What resources are available for more open-ended or nonroutine questions to enable our students to delve beneath the surface and actually think about what they are doing? Well, I can't answer all these questions but here are a few thoughts...


1) Write two examples for which the lcm of two numbers is their product.
2) Write two examples for which the lcm of two numbers is not their product. The numbers in each example must be distinct (different).

3) The lcm of 12 and N is 24.
a) What is the greatest possible integer value of N?
b) What is the least positive integer value of N?


These are just a few samples to start you off. You could probably come up with better ones or you've read some excellent ideas in some publication. Please share...

To see a more challenging version of the examples above, click Read more...



You might want to give the following for homework or an extra practice problem in class. Do you think students will require a calculator? How about telling them they cannot use it!

The lcm of 100 and N is 500. What is the least positive integer value of N?

...Read more

Number Theory, Logic, Proofs and Patterns for Middle School and Beyond...

HAPPY HOLIDAYS!

The following is a series of apparently straightforward arithmetic problems for middle schoolers. However, the objective is to have students justify their reasoning beyond "guess and test" methods. Proving there is only one solution or none requires more careful logic using algebra as needed. Students will need some basic algebra for the "proofs." For the younger student, modify these questions to have them find the squares in questions 1,2,3 and 5. Take this as far as you wish...


In the following, square refers to the square of an integer. Justify your reasoning or prove each of the following.

(1) There is only one square which is 1 more than a prime.

(2) There is only one square which is 4 more than a prime.

(3) There is only one square which is 9 more than a prime.

(4) There is no square which is 16 more than a prime.

(5) There is only one square which is 25 more than a prime.

(6) Can one generalize this or not??

Click Read More for selected answers, solutions...



Selected Answers, Solutions

(2) If n² is 4 more than some prime, p, then we can write
p = n² - 4 = (n-2)(n+2). Since p is prime, the smaller factor must be 1, so
n-2 = 1 or n = 3. Thus, there is only one square, 9, which is 4 more than the prime, 5.

(4) p = n² - 16 = (n-4)(n+4). There n would have to equal 5, n² would equal 25 but 25 - 16 = 9 is not prime.

(6) If there were a general rule would that mean we'd have a formula for primes?

Your thoughts about these questions...

...Read more

Remainders and Number Theory Challenges for Middle School and Beyond

Edit: #4 below has been corrected. I am indebted to one of mathmom's astute students for catching my error!

Number theory is part of many states' standards but usually only at a basic level (factors, multiples, primes, composites, gcf, lcm). Below you will find a problem for your students to work on (preferably with partner). It is not an introductory problem using remainders so they would have needed to do preliminary work beforehand.

Here are some suggestions for developing the foundation for today's challenge problem:

(1 ) List the first 5 positive integers which leave a remainder of 1 when divided by 2? Describe, in general, such positive integers.

(2) List the first 5 positive integers which leave a remainder of 3 when divided by 13? If you subtract 3 from each of these, what do you notice? Explain!

(3) List the first 5 positive integers which leave a remainder of 12 when divided by 13. If you subtract 12 from each of these, what do you notice? If, instead you ADD 1 to each of the 5 positive integers, what do you notice? Explain!

(4) What is the least positive integer N, greater than 1, which leaves a remainder of 1 when divided by 2, 3, 4 or 5? [Ans: 61]
Note: The word 'or' may be confusing or inaccurate here. Modify as needed!

Now for today's challenge (allow use of calculator):

What is the least positive integer which satisfies ALL of the following:
leaves a remainder of 1 when divided by 2
leaves a remainder of 2 when divided by 3
leaves a remainder of 3 when divided by 4
leaves a remainder of 4 when divided by 5
leaves a remainder of 5 when divided by 6
leaves a remainder of 6 when divided by 7
leaves a remainder of 7 when divided by 8
leaves a remainder of 8 when divided by 9.

Notes/Comments
This challenge looks harder than it is. Variations of these often appear on math contests for middle school and beyond. Simpler versions like example (4) above have appeared on the SATs.

Of course, modular arithmetic and congruences would make this problem trivial but that is non-standard and requires more time to develop.

I will not yet post the answer or possible solution...

HOW MANY FASCINATING FACTS can you find about 97? about 153?

Note: There has been a revision in one of the properties of 17 below - I'm sure you already caught the error! Joshua caught another error involving the 4th powers - both have now been corrected.

For middle or high school students:

Number sleuths -- get into your detective groups. No calculators for the first 5 minutes.
You will have a total of 15 minutes to uncover as many interesting or fascinating facts as you can about the number 97.
One member of the group must record these and report back. Make two columns.
The first labeled: Discoveries found without the calculator.
The other column: Discoveries found with the calculator.

Ah but you're wondering what makes some fact interesting or fascinating. That's pretty subjective, right? Let's model one:

17

• Prime number
• Both of its digits are odd
• The product of its digits is prime
  
• Can be written as a sum of 2 consecutive integers: 17 = 9+8
• If its digits are reversed, the resulting number, 71, is also prime and the difference of 71 and 17 is 64 or 8²; oh, and 8 just happens to be 1+7!!
  Note: This is clearly incorrect - 3-point penalty!!
  
• Can be written as a sum of squares: 17 = 4² + 1²
• Can be written as a difference of squares: 17 = 9² - 8²
• Can be written as a sum of a perfect square and a perfect cube: 17 = 3² + 2³
• Can be written as a sum of 2 consecutive fourth powers: 17 = 1⁴ + 2⁴; in fact, 17 is the least integer which can be written as the sum of two distinct nonzero 4th powers
  
• You can get your driver's license in some states on your 17th birthday (this is clearly the only fact that's interesting to an adolescent - I had asked one of my students what's special about March 14 and he replied, "It's my birthday and it's also the Junior Prom.")
  
• 17 is the hypotenuse of a Pythagorean triple; 8-15-17 [high school level?]
  
• 17 is part of another Pythagorean triple: 17-144-145 [high school level?]
  
• Is there any end to this list?

Well, you get the idea.
You will have 15 minutes to complete your investigation of the number 97.

To rate each of your lists, we can assign a point value to each fact. For example, the first four facts listed above for the number 17 could each receive one point. Some of the other facts could receive 2, 3, 4 or even 5 points (max) for being more difficult to find or just more amazing. If any fact is incorrect, 3 points are deducted, so you'd better do independent fact-checking on your team! We will then determine the top 3 lists from this rating system and those teams will receive worldwide recognition by having their results appear on You Tube (just their lists, not names or faces for reasons of confidentiality of course). This will surely go viral in 17 seconds or less. Oh, so you'd rather receive 97 bonus points on your next test?

Oh, I forgot to mention. As a super extra credit bonus project/assignment, do the same this evening for the number 153. You will need to email me with your individual lists by 10 PM (yes, yes, we all know this is unfair to students who do not have access to email or whose parents may not give permission for this or for some other reason, so this is just an option!).

Digit Discoveries and Algebra for Middle School - An Investigation

The results below are well-known but, as usual, I am offering an investigation for the classroom that has many objectives:

(1) Digit properties of multiples of 9 (and their 'proofs')
(2) Review place-value and algebraic representation
(3) Investigate patterns based on data collection
(4) Develop inference and conjecture
(5) Introduce students to algebraic proof
(6) And, of course, practice for those open-ended questions we've all come to know and love...

Children are often fascinated by the discoveries they can make regarding 2- and 3-digit numbers. At some point in middle school all students should either discover on their own or be introduced to the remarkable properties of the number 9 in our base 10 number system. The investigation below will explore some of this.

Students are also fascinated by the results of taking a 2- or 3-digit number and reversing its digits. With or without calculators, students like to see how these numbers are related, particularly when they are added or subtracted. In this activity, they will have the opportunity to discover some of these properties and use basic algebra to explain why they work. Perhaps, this will also lead to questions about palindromes, but that's for another day...

The questions below are designed for middle schoolers through Algebra 1. The proofs require some basic algebra, so you can make those parts optional for the prealgebra group. For this group, having them state their conjectures and suggesting possible explanations are more than enough.

STUDENT/READER ACTIVITY/INVESTIGATION

(1) List all of the 2-digit multiples of 9. What do you notice about the sum of their digits?
(2) Using the fact that any 2-digit number can be represented algebraically as 10a+b, show/justify/explain/demonstrate/prove the following:

If a 2 -digit number is a multiple of 9, so is the sum of its digits AND
if the sum of the digits of a 2-digit number is divisible by 9, then the number is a multiple of 9.

(3) If you made sense of (2), why stop with 2-digit numbers! State and prove a similar result for 3- and 4-digit numbers!

Now for reversals:

(4) To be a mathematical researcher, one needs to do what the scientific researcher does. Collect lots of data first, then make conjectures and PROVE them! Choose at least 5 different 2-digit numbers, in addition to the examples below, and complete the table.

Number..........Reversal............Sum..........Difference (Larger-Smaller)
41.....................14.......................55...............27
33....................33......................66................0
72....................27......................99................45
Your turn - do this FIVE more times.

(5) Make conjectures about the how the sum and difference are related to the digits of the original number. Using the algebraic representation 10a+b for any 2-digit number, PROVE your conjectures (or disprove them!).

(6) 72 and 27 are not only reversals. They are are also both multiples of 9. Does this have to be true for any 2-digit multiple of 9? Explain! Further, is there a special property for the sum of the number and its reversal in this case. Make sure you verify conjectures for several cases before attempting to prove it.

(7) Make a similar table for 3-digit numbers. Is there an obvious relationship for the sum of the number and its reversal this time? The difference? Make conjectures and PROVE them!

If you feel this activity is useful, please comment, share it and rate it below. Enjoy!

Elementary Arithmetic for the New Year? A Partition Problem for Middle Schoolers or for SATs

The following question(s) are appropriate for grades 5-12. How is this possible? Well, the mathematics needed to solve it is elementary! Of course, as with most math problems of this sort, the meaning/interpretation of the question is the challenge for most. Then there is the issue of going beyond the question to look for deeper mathematical meaning...

To help the younger student get started, we will begin with an example and then proceed with the question. Older students might need the same since the language of the question may be vague or difficult to comprehend:

The number 6 can be written as a sum of one or more odd positive integers in exactly 4 ways:
5+1, 3+3, 3+1+1+1, and 1+1+1+1+1+1. As you can see, we are not considering the order of the summands.

Also, 6 can be written as a sum of one or more different positive integers in exactly 4 ways:
6, 5+1, 4+2, 3+2+1. Again, different orders are not included in our list.

Anything of interest yet? Should students naturally raise their hands and ask questions about these two problems or do we have to cue them? At this point, the educator has many options. Here are a couple:

Pair of Problems:
List all ways to write 9 as a sum of one or more odd positive integers. How many ways?
List all ways to write 9 as a sum of one or more different positive integers. How many ways?
What do you notice?
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Investigation (in groups):
Make a table for positive integers from 1 through 10 as follows (I'm abbreviating the column headings). Also, include extra columns for the number of ways for each.

N..........N as a Sum of Odds...............N as a Sum of Different
1.........................1.......................................................1
2.......................1+1....................................................2
3.......
.
.
.
.
10

Comments:
(1) Can you think of at least 3 benefits from having students doing these?
(2) If class time does not permit further investigation, is it worth assigning these for homework, enrichment, extra credit, etc?
(3) Anyone imagine there might be some highly sophisticated ideas from number theory behind these simple questions? Anyone know who posed these kinds of questions originally and solved the general question?

What is the Largest 3-digit Multiple of 7? - A Middle School Activity on Remainders, Multiples and the Division Algorithm

Please don't forget to give proper attribution when using this activity in the classroom (see sidebar).

STUDENT/READER CHALLENGE #1:
What is the largest 3-digit multiple of 7?

Would you expect most students to reach for the calculator and, without much thinking, test 999,998,997, etc., until they reach 994?

This is an opportunity to review the Division Algorithm, the conceptual meaning of remainder and how important this integer can be when solving problems involving multiples, factors, and various number theory questions (not to mention those repeating pattern problems so popular on standardized tests).

From the calculator, students obtain a result like 1000/7 = 142.8571429. The last digit is rounded which disguises the repeating decimal but some will know that. You ask for the remainder from this division problem and you may get an answer like 6/7 or blank stares or some decimal. Because we live in a calculator environment, students may have already acquired ways to obtain the remainder from the calculator display. Older students with their sophisticated graphing calculators may have a function that returns the remainder (like mod) or some application they downloaded for this purpose.

Here's one way students find remainders on their calculator:

Ignore the decimal, that is, look at the greatest integer value of the quotient, namely 142. Then, the remainder can be obtained from: 6 = 1000 - 142⋅7. This is just another form of how students should check their division:
142⋅7 + 6 = 1000

Students often think about this procedurally, not conceptually. In fact, a thorough understanding of remainders and the division algorithm would make the questions in this post fairly simple.

In general, suppose B is divided by A, producing an integer 'quotient' Q and a remainder R, where R is an integer satisfying A > R ≥ 0. Then the Division Algorithm states:
QA + R = B or R = B - QA

As an aside, some students are taught or discover another calculator approach for finding the remainder:
Subtract (discard) the integer part of the decimal, leaving 0.8571429, then multiply this result by the original divisor 7. Like magic, the display reads 6. Students should realize that the calculator internally stores more places than it displays! It may be worth it to demonstrate why this method works or have students investigate this, so I'll make it part of today's challenge:

STUDENT/READER CHALLENGE #2:
Explain why the above procedure works for finding remainders. You may want to employ an algebraic derivation, using A, B, Q and R as above.

Students should now use remainder concepts to solve the original problem and the following:

STUDENT/READER CHALLENGE #3
What is the largest 6-digit multiple of 7? Explain your method carefully.

Note: If time permits (like right before a vacation), you may want to introduce modulo arithmetic (congruences, etc.) to solve the remainder problem in a more compact form:
10 ≡ 3 (mod 7) → 10⁶ ≡ 3⁶ ≡ (3²)³ ≡ 2³ ≡ 1 (mod 7), using the fact that 9 is congruent to 2 modulo 7. Thus, the remainder is 1 and the result follows. Of course, there's a great amount of overhead in developing this much number theory, but if you're looking a holiday challenge...

Does doubling an integer double the number of factors? A Deeper Investigation for Middle School

NOTE: PLS READ THE COMMENTS FOR A DETAILED DISCUSSION OF THIS PROBLEM, WHICH SHOULD PROVE QUITE CHALLENGING FOR MOST MIDDLER SCHOOLERS.

The previous activity I posted regarding integers that have exactly four factors might lead to some interesting discussion regarding a general description of such numbers. All of these kinds of problems could be handled by simply giving students the general rule for determining the number of factors of any positive integer. I have given this well-known number theoretic formulation in earlier posts, so I won't review that at this time. However, there is a greater benefit to be derived from having students investigate these relationships. The following activity should be adapted to meet the needs of your students.

Some educators react to these kinds of deeper investigations with reactions like:
(a) I have a curriculum to cover. I don't have time for this.
(b) Unless this kind of question appears on state testing, it's simply not practical for me to do this.
(c) My students are just not ready for this kind of thinking.
(d) Dave, you're out of the classroom now, so you're forgetting the realities of most classrooms. Some students don't know their basic facts and you want me to do higher-order thinking! Gee, Dave, are you forgetting we have classified children mainstreamed in our classes? Get real!
(e) Dave, stop suggesting HOW we should teach and just give us the problem. You're trying to impose your style on others - it doesn't work - we each bring our own style to a lesson.

[Comment: I have strong reactions to some of the above, but then I'd be arguing with myself! I'll respond to some of these in the comments section or devote an entire post to these critical issues if my readers decide to respond to this.]

I'm certainly not suggesting that these kinds of explorations should BE the curriculum. There must be a balance between these problem-centered approaches and skills development. I am suggesting there needs to be some time devoted to deeper cognitive processes to foster mathematical development. The following investigation is far from one inch deep! I may continue it later but I'm hoping some will suggest extensions, make comments or report back how it played out in real classrooms (also how it was adapted/revised).


INVESTIGATION/READER CHALLENGE

Part I
1. The number 6 has 4 factors: 1,2,3,6 (or in paired form: 1,6;2,3).
Suggested Questions:
If we double the number 6, what do you think will happen to the number of factors? Will it increase or stay the same? Will the number of factors also double?
Mathematicians, like scientists, make conjectures or educated guesses, but not wild guesses! We need some evidence or data on which to base our conjectures. With your partner, fill in (and possibly extend) the following table, then formulate your conjecture using correct mathematical language:
Positive Integer.................Number of Factors
6...........................................4
12.........................................____
24........................................____
48.......................................____

Do you think we have enough data to make a conjecture or should we continue the table? Record your observations and then state your conjecture or 'rule'.

Teacher Tip: Depending on the maturity of the group and their experience with these kinds of formulations, you may want to start them off with a prompt:
If we double a positive integer, the number of factors _______________.

Many students will be convinced they have found a mathematical rule that will always work. That's one of the objectives of this investigation: To help them understand that
(a) Pattern recognition does not a rule prove!
(b) The conjecture is based on starting from the number 6. There is no basis for assuming that their 'rule' will be valid if we start from a different positive integer!

Part II
This time have students start from a different integer: 18
Make a table similar to the one above, again doubling the integer in the left column.
Again, record your observations and then state your conjecture or 'rule'.

Suggested Questions:
Do you think there is a more general rule that covers all cases or are there simply different rules for different integers? If you were going to investigate this further, what other kinds of starting positive integers would you try?

Teacher Tip: Asking many questions stimulates student thinking and leads to more questions and deeper thought processes on their part. A free interchange for a couple of minutes is invaluable here to have students come to see that mathematical research requires persistence and an attitude of inquiry. As Ms. Fribble from the Magic School Bus would say: ASK QUESTIONS! (or something like this!).

Part III - Start from an odd integer this time: 15
Part IV: To be continued...

Middle School or SAT Math Activity - The Four Factors Problem

There are countless problems involving the factors of a positive integer we're seeing in middle school classrooms and on standardized tests these days. They are often used as challenges or warm-ups and questions similar to the one(s) below have appeared frequently on this blog. Students become more proficient with this type of question by doing many variations repeatedly over time. As they mature, they will come to appreciate a more general approach to finding the number of factors of any positive integer. Number theory is now included in most states' standards so there needs to be some time devoted to this topic on a regular basis.

STUDENT PROBLEM/READER CHALLENGE

This problem/activity is often best implemented in small groups. Each member of the group should make their own list and then compare, however, they might want to divide the labor by having some students do the numbers up to 50 and others do the rest.
Suggested Time for Activity: 15-20 minutes (the problem can be explored further for homework or a challenge, then revisited the following day for 5 minutes).

The number 12 has 6 positive integer factors: 1,12;2,6;3,4.
(a) List all positive integers up to and including 100 that have exactly four factors.
(b) Higher-order: These numbers fall into 2 categories. Describe these categories.

Alternate Problem (shorter time needed): What is the largest 2-digit positive integer that has exactly 4 factors?

Products of Digits: Challenges for Everyone...

[You may also want to look at the preview of the interview with Alec Klein, author of A Class Apart, to be hosted on MathNotations. Alec has agreed to answer my questions about Stuyvesant HS in NYC, other specialized schools and gifted education.]


One never knows where the inspiration for a math challenge might come from. This one came from a book I devoured recently, entitled The Righteous Men by Sam Bourne. If you enjoyed the Da Vinci Code and solving the number puzzles and other codes embedded in the story, you will definitely enjoy this novel. One of the most important clues to unraveling the mystery in the story was stated as a riddle:
Just men we are, our number few
Describable in digits two
We're halved if these do multiply
If we few perish then all must die.

Ok, now some of you are going to ignore the math problem and run out to find this book, but for those who would like a more prosaic version of the math problem, here is our first conundrum:
Note: Students from middle school on can attempt some of these although the proofs are a reach.

(1) Determine a 2-digit positive integer the product of whose digits is one-half the integer.
Now, it won't take you long to find such a number (once you get past the elliptical phrasing), but that's just to whet your appetite. The real challenge begins:

(2) Prove that your answer to (1) is unique, i.e., there is only one solution to the problem.

Comment: We're looking for more than an exhaustive search through all ninety 2-digit numbers or a programmed solution. The key to this and all of the remaining questions is to find an approach to solving a single equation which has 2 or more variables whose domain is the set of positive integers. Students are usually not introduced to solving such equations but they appear frequently on SATs and Math Contests. Because we are looking only for positive integer solutions, a standard algebraic approach must be supplemented with arithmetic concepts and testing of several possibilities. Number theorists refer to these as Diophantine equations.

(3) How about 3-digit integers? If 12 were permitted as the hundreds' digit, then (12)96 would satisfy the problem since (12)(9)(6) = 648, which is one-half of 1296. Unfortunately we can't allow that, so your challenge is to prove that there are NO 3-digit numbers (meaning positive integers) the product of whose digits is one-half the number.

Comments: Again, you're confronted with an equation but this time there are 3 variables. One approach is to solve for h in terms of t and u (I'll let you guess the symbolism) but there surely must be numerous possibilities! Or are there...

(4) Since we couldn't find a 3-digit number with the property, we'll rework the ratio:
Determine a 3-digit number the product of whose digits is one-fourth the number.

(5) Of course, I can't let you off the hook that easily. Prove that your solution in (4) is unique!

Standard disclaimer: These results have not been independently verified. Translation: I devised problems (2)-(5) and therefore there could be errors. If you find other solutions, let me know!

Reminder: Don't forget to give Proper Attribution when using these original problems. See the instructions in the sidebar. I appreciate your understanding and acknowledgment.

More Prime Searches...

[Update - The answers to the questions below appear in the comments section.]

I am continually amazed by some of the search phrases that lead to this blog. Although many are math topics about which readers are looking for more information, some are actual math problems that intrigue me. Here is one for today that led me to probe more deeply. On the surface it doesn't seems to require that much analysis, just an understanding of the rule for finding the number of factors of a positive integer from its prime factorization (see my earlier post on Fun with Factoring) and a good list of primes, but you may see something deeper here. At the least, it looks like an interesting investigation for middle schoolers and beyond with a webquest built in. I am indebted to the searcher whoever she or he may be!

Here's the actual search phrase I found:

1. What is the largest 3-digit integer with exactly four factors?

Before revealing the answer, I decided to expand this a bit:

2. (Easier but still worth doing) What are the largest and smallest 2-digit integers with exactly four factors?

3. Ok, so naturally, we would also ask: What is the smallest 3-digit integer with exactly four factors?

5. Keep going... What are the largest and smallest 4-digit integers with exactly four factors?

Of course, a simple factoring program written on a graphing calculator or in C++, etc., would suffice, but see how long it takes you to search and how logical reasoning and analysis can save some time. Of course it always helps to have a list of primes handy so don't forget the ultimate primes list from the U. of Utah.

Before you decide this is just a way to keep kids busy, try it. If you see a pattern or wish to expand this, go for it!

A Number Theory Problem for 2-12-07: Mission Impossible?

PLS READ THE COMMENTS TO SEE HOW THE LESSON FARED OVER TWO DAYS. DO YOU THINK I HAD TO MODIFY IT A LITTLE OR A LOT TO MAINTAIN THEIR INTEREST?

Thanks to jd2718 for motivating me to include the following famous number-theory conundrum. You can find many references to it on the web but I remember it from the classic text by Hardy and Wright. Give this as a challenge bonus problem or as an activity for your middle or high schoolers. Revise and modify it to make it appropriate for your students. I've tried to turn this into something students can at least attempt, but you could probably make it much more user-friendly...
Remember: My goal is to provide meaningful activities for ALL of your students, not only for honors or accelerated classes. The challenge is to modify them for students who struggle in math. The real test comes when I implement today's problem in my 'skills' class to which I frequently refer. That is my plan -- I'll let you know if it worked or was a disaster! Pls note that these activities also provide practice for those open-ended types of questions that now frequently appear on standardized and state assessments.
Finally, I hope you enjoy these challenges, but my primary target audience is students. I am really interested in reading students' reactions to these.

MISSION IMPOSSIBLE?
Now, boys and girls, we know that, despite all attempts by the most famous mathematicians, no one has yet devised a formula for primes. At one time, it was thought that 2^p-1 would always be prime if p is prime:
2^2-1 = 3; 2^3-1 = 7; 2^5-1 = 31; 2^7-1 = 127 -- all primes. But alas, 2^11-1 - 2047 = (23)(89). Imagine the disappointment!
But I think I found a method that will produce primes EVERY time! If you can do all parts below and prove I'm wrong, you get 5 bonus points. You'll get at least one point for doing part (a).

Ok, I decided to start with my favorite prime 41, since that's the age I once was!
41 + 2 = 43
43 + 4 = 47
47 + 6 = 53
53 + 8 = 61
61 + 10 = 71
71 + 12 = 83
...
Here's your mission:
(a) I listed the first 6 and they're all prime. You can see the pattern, right? Keep the sequence going until you reach 40 numbers. Check that they are all prime by researching a list of primes on the web. So, am I right? Am I famous now?
(b) My method is easy to see but harder to describe algebraically. Here's one way:
To get each number in the list after the first, you can see that I added the next even number to the preceding term. SHOW that the following recursive description produces the first 6 terms:
a(1) = 41; a(n+1) = a(n) + 2n, n = 1,2,3,... where the notation a(n) refers to the nth term. Then explain why this will generate all the terms.
(c) This one is harder but I'll give you a hint: Devise a polynomial that will generate this sequence of numbers when n = 1,2,3,....
Hint: Think of a quadratic like n^2 +.... Just remember, when n is replaced by 1 it has to yield a value of 41!
(d) Show that my amazing sequence 'blows up' when you reach the 41st term. Why? There goes my million dollars from the Clay Institute!
Extension: Is there any other sequence like this? What's special about 41? Happy web-questing...