SAT CHALLENGE - ODD NUMBERS OF FACTORS

How many positive integers less than 1000 have exactly

(a) 3 positive integer factors
Ans: 11

(b) 5 pos int factors
Ans: 3

(c) 7 factors
Ans: 2

Is this topic in the middle school core standards? Under divisibility? Factors?

Have you seen questions like these on state tests? SATs?

What strategy would you like your 6th-8th graders to use? Assuming they don't know a 'rule' for this problem, how can they best discover a pattern? Would it make sense for students to make a 2-column table of integers and number of factors?

Why am I addressing middle school curriculum when the title of this post refers to SATs?

Is this question not worth all the time it would consume?

Do you believe this question is only for the 'mathletes' who take math contests?



Sent from my Verizon Wireless 4GLTE Phone

The Largest Odd Factor of 90? Too Easy? How About A Million? A Googol!

Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:

PRINCE? NAH! E-ROI!

We would hope that by grade 5, most youngsters would be able to answer the first question in the title fairly rapidly and without a calculator. Or are you thinking many would incorrectly blurt out '9' as the answer?

Well, why should anyone care about finding the largest odd factor of some positive integer? Will it lead to a better understanding of the origins of the universe? Perhaps not, but these questions may deepen student understanding of

(a) Factors
(b) Prime factorization (particularly of powers of 10)
(c) The important concept that an even number may have odd factors but an odd number can never have any even factors!
(d) Other ideas...

Ninety is not a very large number so students will usually see that the answer is 45. However there are so many ways of looking at this simple result. So many important methods -- so little time! Further, a method that works effectively for 90 may not be as effective for 1,000,000 or a googol.

My suggestion is to give middle schoolers the 'million' problem, let them work with a partner, allow the use of a calculator and see what happens.


Here are some thoughts:

(a) Which of the following is more instructive, more important conceptually?

Writing 1,000,000 as 2⁶⋅5⁶, etc.,
OR
Having students, on the calculator, divide 1,000,000 by 2, then the quotient by 2 and so on, until an odd result occurs

(b) Do these 2 approaches reinforce/develop the same concepts/skills or different ideas?

(c) Which method is most reasonable for 90? for 1,000,00? for a googol?

(d) Does the calculator enhance or not enhance understanding here? Does it depend on the number we start with?

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?

Taking the 'Unsummable' Numbers to a higher level: An Algebraic Proof

[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.]


Please comply with the 'Proper Attribution' statement that now appears in the sidebar.

Anyone recall a detailed investigation I posted in Feb '07 about numbers which can or cannot be written as a sum of 2 or more consecutive positive integers? You will probably want to quickly review that for this discussion. That investigation was implemented in a 9th grade prealgebra class with students who had struggled with math for a long time. They worked for the entire period (and into the next class as well) and expressed satisfaction and a sense of accomplishment. One student, KC, even found a way to express those numbers which were unsummable!

Today, we will take this question of which numbers are unsummable to a higher level. The challenge for my readers and for students is to use methods from Algebra 2 and basic number theory (primes, factors) to prove a conjecture made by one of my former students.

Quick background:
Students were asked to investigate those positive integers which can be written as a sum of 2 or more consecutive positive integers. I started them off with examples like
3 = 1+2; 5 = 2+3; 6 = 1+2+3, etc.
They worked in pairs and completed a table up to 36 over a couple of days. Most quickly realized that every odd positive integer starting with 3 could be represented but not every even positive integer. Working in pairs helped students to catch common arithmetic/logic errors and the results were reviewed after every 10 numbers or so in order to insure that all students had accurate results to work with.

Here's your challenge for today:
Rather than demonstrate which numbers can be represented as such a sum, your mission is to prove the following:
Powers of 2 are unsummable, i.e., a power of 2 can never be represented as a sum of 2 or more consecutive positive integers.
Notes:
(1) The whole notion of algebraic proof may be new for some students, so this may require some demonstration first.
(2) Students will need to know the formula for the sum of an arithmetic series, so this challenge would be appropriate after learning or reviewing that. The instructor however could develop that formula earlier on or simply provide the formula.
(3) Some understanding of the Fundamental Theorem of Arithmetic is needed here, i.e., every positive integer is either prime or can be written as a product of primes in a unique fashion. This theorem often goes unmentioned or taken for granted in middle school - time to bring it back?
(4) Some readers may find a way to prove this without using the algebraic formula mentioned above. Share that as well!

Enjoy...

Helping Students Think 'Outside the Box'...

[Update: I'm adding an additional problem at the end. This question seems to be of interest to some since I've seen it in a Google search for awhile now. Answers and solutions to some of the problems will shortly appear in the comments.]


The issue for me as a math educator has always been:
How do we enable children to think conceptually?

Here are some standardized types of questions each of which can be solved by a variety of methods.

In each of the problems below, there is a conceptual approach that requires skill, knowledge and some insight. We all know as educators we can do something about the first two (provided students are given enough practice and they do it!), but how do we develop insight? I can only tell you how my insight improves: When I tackle harder problems or those requiring me to 'think differently'. I am sure there are those out there who are able to invent these methods on their own, but, as for me, I have to work at it and think about it!

The first 3 questions involve ratios. Since we know how the current generation feels about fractions, these may cause students to feel some frustration!

See if you can find an 'insightful' or conceptual approach. Also, ask yourself how most middle or secondary students would approach these:

1. Background Terminology: For those of the younger generation who may not have heard of the term proper fraction, it means a ratio of positive integers in which the denominator is larger than the numerator. Thus, 4/3 and 3/3 are improper; 2/3 and 1/3 are proper. Also, the phrase 'in lowest terms' means that the greatest common factor of the numerator and denominator is 1. (But everyone knew that, of course!)

If 10/n, 14/n and 15/n are proper fractions in lowest terms, what is the least possible positive integer value of n which is not prime?

2. For how many integer values of n is 11/n between 1/9 and 1/10?
Note: Is this an algebra problem? A guess-test 'plug-in' problem? A calculator problem? Or just a fraction 'exercise'?

3. Fahrenheit temperature is related to Celsius by the equation: F = (9/5)C + 32.
An increase of 36 degree F. is equivalent to an increase of how many degrees Celsius?
Note: Many students struggle with an approach here. Some try it algebraically, most plug in some initial temperature, virtually none I have observed think conceptually about the meaning of ratios.

4. A cube 6 inches on each edge is sliced 'horizontally' to form 2 congruent rectangular solids. If these 2 solids are joined to form a rectangular solid which is not a cube, the surface area of this resulting solid is how many more square inches than the surface area of the original cube?
Note: Some youngsters simply 'see' this with little or no calculation! I guess you could say, they really 'think outside the box!' (sorry 'bout that...)

5. What is the smallest positive integer having exactly seven factors?

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!

FUN with FACTORING: SAT-Type and Contest-Type Problems on Factors

[Important Note: Thanks to 'e', Question 1 below has been modified to 'five' 2-digit numbers with 12 factors. Please read the comments for follow-up.]

[The ideas and problems today are suitable for grades 6-12.]
Those into number theory know many basic principles that help them solve problems involving factors that seem arduous at first. One extremely useful formula that mathletes are taught early on and SAT students should know is the following (the abstract form hides how easy it is, so get to the example quickly!):

The FUN-damental FACTORING RULE: (I coined this silly name so don't quote me!!):
If the positive integer N = p₁^e₁p₂^e₂p₃^e₃...p_n^e_n then
N has (e₁+1)(e₂+1)(e₃+1)...(e_n+1) positive integer factors!

The p_i's are distinct primes and the e_i's are positive integer exponents.
Note: From this point on, whenever the term factor is used, it refers to a positive integer factor.

Ok, we need an example fast!

Example: How many positive integer factors does 48 have?
Solution: First we need to write the prime factorization of 48 = 2⁴3¹
[For larger numbers, writing the prime factorization is more problematic and a computer program or a calculator like the TI-89 would be useful].
ADD ONE to each of the exponents and MULTIPLY: (4+1)(1+1) = 5 x 2 = 10. Voila!
Verify: 1,48; 2,24; 3,16; 4,12; 6,8. Ten, indeed!

The explanation of this very handy rule involves some basic combinatorial thinking since EVERY factor of 48 (similar argument for N, in general) can be written in the form 2^a3^b where a could be 0,1,2,3, or 4 and b could be 0 or 1. Thus, there are FIVE (4+1) possibilities for a and TWO (1+1) possibilities for b. By the multiplication principle, there would be 5 x 2 ways to form different factors of 48.

Ok, so here are some examples (not very challenging) for middle schoolers and on up:

1) Using our FUN-damental Rule above, find the five 2-digit positive integers which have exactly TWELVE distinct factors.
The object is not to list every number from 10-99 and count factors!
Extras: Explain why a 2-digit number cannot have more than 12 factors. What would be the smallest integer that has more than 12 factors?

2) How many factors does 2007 have?
[Easy, once you have the prime factors, but it's always fun finding them for each new year or showing it is prime. Students better know why 2007 is NOT prime!!].

3) SAT-type (easy using above rule): If N = pqr, where p, q and r are distinct primes, explain, without listing or plugging in numbers, why N has exactly eight factors.
Then list the eight factors in terms of p, q and r.

There are endless variations and applications of the FUN-damental Rule. I'll leave it my readers to suggest really 'wicked' ones!