Latest CAT Aptitude Question SOLUTION: 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

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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

Asked In CAT shyamjee kumar (11 years ago)

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ravindra kumar  (11 years ago)

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.

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