Algebra 2

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Two distinct numbers a and b are chosen randomly from the set {2, 2², 2³, 2⁴, ......, 2²⁵}. Find the probability that log_ab is an integer.

#Mathematics #Algebra #Probability

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5 Answers

591

Akshay Ginodia ·2014-03-07 00:30:25

31/150

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1133

Sourish Ghosh ·2014-03-07 22:11:24

No. of ways of selecting a and b is ²⁵C₂.2! = 600
No. of ways in which a and b can be chosen such that logbloga is an integer:
[251] - 1 + [252] - 1 + [253] - 1 + [254] - 1 + .... + [2525] - 1 = 62, where [.] is the floor function.
Probability = 62/600 = 31/300

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Akshay Ginodia opps! didnt apply permutations to a&b
Upvote·0· Reply ·2014-03-07 22:46:54
Akshay Ginodia Btw can u plz explain ur solution.? :)
Upvote·0· Reply ·2014-03-07 22:47:54
Sourish Ghosh If you choose any number 'k' b/w 1 and 25 then no. of multiples of k <= 25 is [25/k] - 1. For example take 6. No. of multiples of 6 which is less than 25 is 3 {12, 18, 24} which is [25/6] - 1.
Upvote·0· Reply ·2014-03-14 10:10:58
Aditya Agarwal why did you not consider 6 as well? i mean, if a=6 and, b=6 as well, then also the value of log will be an integer(=1).
Upvote·0· Reply ·2014-03-15 04:34:01
Shaswata Roy @Aditya:It has been mentioned in the question that a and b are distinct.
Upvote·0· Reply ·2014-03-16 21:42:53
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865

Soumyadeep Basu ·2014-03-07 09:25:48

Yes, 31150.

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Manish Shankar ·2014-03-07 21:45:50

Answer is 31/300

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Akshay Ginodia How sir???
Upvote·0· Reply ·2014-03-07 22:08:23
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Aditya Agarwal ·2014-03-13 03:59:55

Why are we permuting?

for eg. if a = 8 and b = 64
log_ab=2 but for the other case where a=64,b=8 value of the log becomes 0.5?

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Akshay Ginodia we are permuting to get the total number of cases
Upvote·0· Reply ·2014-03-14 03:39:01
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