Tuesday, February 12, 2013

What is the smallest positive odd integer which has exactly 10 factors?


Occasionally I like to respond to the topics in the Google searches which bring my readers to MathNotations.

Today's problem in the title of this post
(Humorous speech-to-text aside: "this post" was interpreted" as "disposed")
is a classic math challenge question, difficult SAT- or CCSSM-type question which is appropriate for grades 5-11.

What is the smallest positive odd integer which has exactly 10 factors?
Explain your method.

Answer at bottom of post after shameless promotion...
(Aside: Wouldn't Shameless be a a cool title for a premium cable TV show about a deadbeat dad mathematician!)

REFLECTIONS for my colleagues...
(Wouldn't it be awesome if someone actually read these!)

1.  How many of these types of questions have you seen in textbooks, math contests, SAT's, standardized tests or on other blogs?

2. How different would this question be if the word "odd" were removed? An easier or harder question in your opinion?
(Humorous speech-to-text aside: "the word odd" was interpreted as "The Word of God")

3. Would you like to share some math strategies you have used for this type of problem? Are there instructional strategies you prefer for this? Do you see these 2 questions as equivalent?

4.  Do we have to be the ones to devise variations on this?
My feeling is that all learners, including us, become more proficient at problem-solving and develop deeper understanding when we are asked to pose our own problems!

Do you think your students would, in groups or alone, arrive at variations like "use even in place of odd or drop the word completely? There's only one way to find out!  Perhaps you can share your experiences here...

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Answer to today's problem: 3^4 x 5 = 405  [Correction thanks to Novotny!!]


notovny said...

I'm getting 405 as a tenative answer:

The factors of 405 are 1, 3, 5, 9, 15, 27, 45, 81, 135, 405.

As for how, I started by realizing any odd number would have only odd factors, and started looking at numbers whose prime factorizations used the smallest odd primes. 243 (3*3*3*3*3) didn't have enough factors, 405 did (3*3*3*3*5).

notovny said...

I'm getting 405 as the answer. 405 has the factors
1, 3, 5, 9, 15, 27, 45, 81, 135, 405.

Tried out numbers that had exclusively-odd prime factorizations. 3^5 didn't have enough factors, 3^4 * 5 did.

Dave Marain said...

Thanks, Novotny!!
I credited you with catching my careless error. I'm fortunate to have readers who are smarter and more careful than I am...

The more advanced approach here is to use the number-theoiretic approach:
If the prime factorization of N is
N = (p_1^e_1)(p_2^e_2)...(p_k^e_k), then the number of of positive integer factors of N is
(e_1+1)(e_2+1)...(e_k+1), provable by combinatorics.

Of course, I carelessly overlooked the (5)(2) combination which comes from (3^4)(5^1). Shame on me!!

David Williams said...

What is the smallest positive integer with 2013 factors?


David Williams said...

What is the smallest positive integer with exactly 2013 factors?

Dave Marain said...

David,Great question!
Since 2013=3×11×61,
I would think it's
Agree? You intentionally dropped 'odd'from your question, yes?