+ Reply to Thread
Results 1 to 10 of 10

Thread: Odd divisors

  1. #1
    Eager! pintogajju has disabled reputation
    Join Date
    Jan 2007
    Posts
    87
    Rep Power
    4

    Odd divisors

    What is the number of Odd divisors of 20! ?

    A. 42984
    B. 34560
    C. 2160
    D. 720
    E. 64

  2. #2
    lsr
    lsr is offline
    TestMagic Guru-in-Training lsr 's dreams are becoming reality. lsr's Avatar
    Join Date
    Jan 2005
    Location
    Toronto
    Posts
    896
    Rep Power
    9
    There are seven odd prime numbers between 1 and 20 inclusive.

    3, 5, 7, 11, 13, 17, 19

    To find the highest power of each:

    [20/3] + [20/9] = 8
    [20/5] = 4
    [20/7] = 2
    [20/11] = 1 (13, 17, and 19 also have a highest power of 1)

    9*5*3*(2^4) = 2160

    Answer is C.
    Last edited by lsr; 10-30-2007 at 03:46 AM.

  3. #3
    Within my grasp! vbhup2 just joined TestMagic.
    Join Date
    Oct 2007
    Posts
    161
    Rep Power
    3
    Thanks lsr. Didn't know this method of finding the highest power for a prime factor.

    Can you generalize this rule?

  4. #4
    Within my grasp! Name User just joined TestMagic.
    Join Date
    Jul 2007
    Posts
    259
    Rep Power
    5
    awesome...i love this strategy!

  5. #5
    Within my grasp! pandeyrav is on the way! pandeyrav's Avatar
    Join Date
    Mar 2006
    Location
    Atlanta
    Posts
    282
    Rep Power
    6
    IMO it should be 9c1+9c2+9c3+9c4+.....+9c9 (although no choice looks close, so this in all probabilities is incorrect) because there are 9 odd numbers between 1 and 20 and any combination of these numbers when multiplied will yield an odd divisor. The question does not say prime divisors. I could not understand lsr's solution, but it sure looks very short. i would be interested in knowing its details.

  6. #6
    Eager! samdamn just joined TestMagic.
    Join Date
    Jul 2007
    Posts
    95
    Rep Power
    4
    how abt this strategy for getting to the answer choice (but not know what the actual answer is)....

    primes
    3,5,7,11,13,17,19

    answer must be less than 7! any answer close to that is the right choice!! Hence (c)

    do note that if the answer choices were not that widely ranged, you must follow a different strategy....

    Let me know if you like this strategy?

  7. #7
    Eager! samdamn just joined TestMagic.
    Join Date
    Jul 2007
    Posts
    95
    Rep Power
    4
    pandeyrav, you cannot do what you suggest for odd numbers but for prime numbers only...

    remember, 3,5 are both prime divisors which via combination yield 15 another odd (though not prime) divisor...

    hence, by your solution, the answer would be
    [7C1+....+7C7 + 1]/2 (dont forget 1 as it is also an odd divisor)

    but this results in 128/2 odd divisors...

    hence the answer (E)

    now, I think what we need is the OA?

  8. #8
    Within my grasp! narinder82 just joined TestMagic.
    Join Date
    Sep 2007
    Posts
    174
    Rep Power
    3
    Guys i can Explain the Logic. We have to find out Number of Odd factors of 20!. So Let us start by Expressing 20! in terms of Prime factors. Since

    N = 20 X 19 X18...................1 = 2^m + 3^n + 5^p + 7^q. Our First goal is to find the Highest Power of 2, 3, 5, 7, 11, 17, 19 which divide 20!. Now Here is the Method Which ISR has shown, but in a complex form.

    Highest Power of 2 Which will divide 20! is = 20/2 + 20/4 + 20/8 + 20/16

    = 10 + 5 + 2 + 1 = 18. Remember only Quotients need to be considered

    By similar method the highest Power of 3, 5, 7,11,13,17,19 is 8 ,4,2,1,1,1,1 respectively. There fore now we can express 20! as

    20! = 2^18 + 3^8 + 5^4 + 7^2 + 11^1 + 13^1 + 17^1 +19^1. Since We have to find the odd divisors we will neglect powers of 2.

    Total no of odd Divisors is (8+1) x ( 4+1) x ( 2+1) X (1+1) x(1+1) x(1+1)x(1+1) = 9 x 5x3x16 = 2160 Ans

    I guess it is clear now.

  9. #9
    Eager! samdamn just joined TestMagic.
    Join Date
    Jul 2007
    Posts
    95
    Rep Power
    4
    good explanation... but why (8+1) x ( 4+1) x ( 2+1) X (1+1) x(1+1) x(1+1)x(1+1) ??

  10. #10
    Within my grasp! vbhup2 just joined TestMagic.
    Join Date
    Oct 2007
    Posts
    161
    Rep Power
    3
    If you are asking about why the + 1 when multiplying:

    Because the 3 ^ 0 should also be counted in determining the number of factors. Similarly, the powers for other primes have a +1.

    If the question is about why they are multiplied:

    To determine the total no. of distinct factors (ways they can be combined) that can be formed from the given set of prime factors (like a combination problem).

+ Reply to Thread

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

     

Similar Threads

  1. DS divisors x^2
    By bmwhype in forum GMAT Data Sufficiency
    Replies: 8
    Last Post: 07-06-2008, 05:04 AM
  2. How many divisors does k^2 + mk have?
    By velman in forum GMAT Data Sufficiency
    Replies: 4
    Last Post: 10-27-2007, 11:30 PM
  3. Divisors...
    By r4rohini in forum GMAT Problem Solving
    Replies: 7
    Last Post: 10-06-2006, 04:03 PM
  4. Even divisors from OG
    By mersh in forum GMAT Problem Solving
    Replies: 5
    Last Post: 08-24-2006, 08:29 AM
  5. Divisors
    By trinity0612 in forum GMAT Problem Solving
    Replies: 1
    Last Post: 05-24-2005, 05:47 AM

Bookmarks

What you can do

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts

SEO by vBSEO 3.5.0 RC2