Results 1 to 2 of 2

Math Help - find n such that T(n) = 16

  1. #1
    Junior Member
    Joined
    Feb 2009
    Posts
    28

    find n such that T(n) = 16

    Find the smallest positive integer that has exactly 16 positive divisors.

    I kinda know how to approach this problem but I'm not sure how to answer it in a formal way.

    Please help!!! Thank you in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,407
    Let n = \prod_{i=1}^k p_i^{e_i} be the prime factorization of n.

    If you recall the properties of \tau (n), we want: \tau (n) = \prod_{i=1}^k \left(e_i+1\right) = 16

    Now, 16 = 2^4 can be written as a product of integers in only 5 ways:
    • 2^4 = 16
    • 2^3 \cdot 2 = 8 \cdot 2
    • 2^2 \cdot 2^2 = 4 \cdot 4
    • 2^2 \cdot 2 \cdot 2 = 4 \cdot 2 \cdot 2
    • 2 \cdot 2 \cdot 2 \cdot 2


    So our possibilities:
    • \tau (n) = e_1 + 1 = 16 \ \Rightarrow \ e_1 = 15
    • \tau (n) = (e_1 + 1)(e_2 + 1) = (8)(2) \ \Rightarrow \ e_1 = 7, \ e_2 = 2
    • \tau (n) = (e_1 + 1)(e_2 + 1) = (4)(4) \ \Rightarrow \ e_1 = 3, \ e_2 = 3
    • \tau (n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) = (4)(2)(2) \ \Rightarrow \ e_1 = 3, \ e_2 = 1, \ e_3 = 1
    • \tau (n) = (e_1 + 1)(e_2 + 1)(e_3 + 1)(e_4 + 1) = (2)(2)(2)(2) \ \Rightarrow \ e_1 = 1, \ e_2 = 1, \ e_3 = 1, \ e_4 = 1


    Since we want the smallest integer, we consider the smallest primes in the factorization of n. So this narrows our list down to:
    n = 2^{15} \quad 2^7 \cdot 3^1\quad 2^3 \cdot 3^3\quad 2^3 \cdot 3^1 \cdot 5^1\quad 2 \cdot 3 \cdot 5 \cdot 7
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 8
    Last Post: March 22nd 2011, 04:57 PM
  2. Replies: 2
    Last Post: July 5th 2010, 08:48 PM
  3. Replies: 1
    Last Post: February 17th 2010, 03:58 PM
  4. Replies: 0
    Last Post: June 16th 2009, 12:43 PM
  5. Replies: 2
    Last Post: April 6th 2009, 08:57 PM

Search Tags


/mathhelpforum @mathhelpforum