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Question

Let a1,a2,a3, be a G.P. with a1=a and common ratio r, where a and r positive integers, then the number of ordered pairs (a, r) such that 12r=1log8ar=2010 is
(correct answer + 1, wrong answer - 0.25)

A
40
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B
42
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C
44
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D
46
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Solution

The correct option is D 46
Given: 12r=1log8ar=2010
log8a1+log8a2+log8a3++log8a12=2010
log8[a1a2a3a12]=2010
a.ar.ar2.ar3ar11=82010
a12r66=26030
(a2r11)6=(21005)6
a2r11=21005

Let a=2α, r=2β, where α,β are non negative integers
2α+11β=1005
If α=0, then
β=[100511]=91
β91
Also, 11β=10052α
which is odd, so β is also odd.
β=1,3,5,7,...,91, which are 46.

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