Sandy Wood
Brain Game: A Factor Fact
by Sandy Wood - December 30, 2008 - 7:30 AM

bloghead_braingames.jpg

Factors are the integers that can be evenly divided into a given number. The factors of the number 4, for instance, are 1, 2, and 4, because each of those numbers can be divided into 4 without leaving a remainder. The factors for 5 are 1 and 5, since no other integers can be evenly divided into 5. The factors for 6 are 1, 2, 3, and 6. With that in mind, and ignoring negative numbers…

What is the smallest number
that has EIGHT different factors?

Here’s the SOLUTION.

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Comments (8)

  1. go go 24.

    posted by Jack on 12-30-2008 at 7:43 am

  2. I knew it was 24 without having to think about it…then again I just finished teaching factors to my 4th graders!

    posted by jen on 12-30-2008 at 10:27 am

  3. Forgot the reCaptcha. I apologize if my first comment shows anwyay:

    Actually figured it out! I knew it couldn’t have been a prime number, else there’d be an odd number of factors. I let x be the solution, and looked at pairs of factors: 1 and x are two of the factors, I assumed 2 and x/2 were 2 more factors, 3 and x/3, and finally 4 and x/4. What is x? The least common denominator of x/2, x/3, and x/4 = 24. :) My math degree is finally paying off.

    posted by Eric on 12-30-2008 at 11:16 am

  4. Eric,

    Perhaps a typo. Squares have an odd number of factors. Primes have two factors.

    Your solution was much more elegant than mine. I used Excel to count factors. It works for small numbers, but would be awkward if they asked for the smallest number with 30 factors.

    posted by Bryan on 12-30-2008 at 12:55 pm

  5. Doh, I meant square numbers anyway, not prime.

    posted by Eric on 12-30-2008 at 1:57 pm

  6. Eric,

    Not to get too ticky-tacky, but isn’t the least common denominator of x/2, x/3, and x/4 “12″?

    That being said, the lowest common denominator of x/2, x/3 and x/4 that does not result in a quotient of {2,3,4} when divided by {2,3,4} is 24.

    posted by Matt on 12-30-2008 at 5:19 pm

  7. I learned this little caveat early on: Make sure you ask for a positive number. Otherwise, the correct answer would be -6 (Factors are -6, -3, -2, -1, 1, 2, 3, and 6).

    posted by Michael on 1-1-2009 at 6:05 am

  8. Scratch that–it would be even lower than -6. It would be the lowest negative number with 8 factors, whatever that might be. *is embarrassed*

    posted by Michael on 1-1-2009 at 6:07 am

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