The sum of base-10 logarithms of divisors of 10^n is 792. what is n?
Possible Answers
Selected Possible Answer


12


13


10


11


14

• Because the prime factorization of 10 is 2 ¢ 5, the positive
  divisors of 10
  n
  are the numbers 2
  a
  ¢ 5
  b
  with 0 · a · n and 0 · b · n.
  Thus
  792 =
  Xn
  a=0
  Xn
  b=0
  log10
  ¡
  2
  a
  5
  b
  ¢
  =
  Xn
  a=0
  Xn
  b=0
  (a log10 2 + b log10 5)
  =
  Xn
  b=0
  Xn
  a=0
  (a log10 2) +
  Xn
  a=0
  Xn
  b=0
  (b log10 5)
  = (n + 1) (log10 2)
  Xn
  a=0
  a + (n + 1) (log10 5)
  Xn
  b=0
  b
  = (n + 1) (log10 2 + log10 5)
  µ
  1
  2
  n(n + 1)
  ¶
  =
  1
  2
  n(n + 1)
  2
  (log10 10) =
  1
  2
  n(n + 1)
  2
  :
  Hence n(n + 1)
  2 = 2 ¢ 792 = 2 ¢ 11 ¢ 72 = 11 ¢ 12
  2
  , so n = 11.
• 12 years agoHelpfull: Yes(0) No(2)