ProbabilityHardBloom L3
Question
Two distinct numbers a and b are selected randomly from the set $\left\{ 2,2^{2},2^{3},\ldots.,2^{25} \right\}$. then the probability that $\log_{a}b$ is an integer is equal to
Options
A.$\frac{1}{10}$
B.$\frac{31}{300}$
C.$\frac{8}{75}$
D.$\frac{11}{100}$
Solution
Sol. [Key Concept: Two distinct $a,b$ chosen: probability of the divisibility/relation condition.]
$$\log_{a}b = \log_{2^{m}}2^{n} = \frac{n}{m}$$
| Let | $m = 1$, | $$n = 2,3,\ldots\ldots,25$$ | number of ( $m,n$ ) $= 24$ |
|---|---|---|---|
| $m = 2$, | $$n = 4,6,\ldots\ldots,24$$ | number of ( $m,n$ ) = 11 | |
| $m = 3$, | $$n = 6,9,\ldots\ldots,24$$ | number of (m, n) = 7 | |
| $m = 4$, | $$n = 8,12,\ldots\ldots.,24$$ | number of $(m,n) = 5$ | |
| $m = 5$, | $$n = 10,15,20,25$$ | number of $(m,n) = 4$ | |
| $m = 6$, | $$n = 12,18,24$$ | number of ( $m,n$ ) = 3 | |
| $m = 7$, | $$n = 14,21$$ | number of (m, n) = 2 | |
| $m = 8$, | $$n = 16,24$$ | number of ( $m,n$ ) = 2 | |
| $m = 9$, | n = 18 | number of (m, n) = 1 | |
| $m = 10$, | n = 20 | number of (m, n) = 1 | |
| $m = 11$, | $$n = 22$$ | number of (m, n) = 1 | |
| $m = 12$, | $$n = 24$$ | number of (m, n) = 1 |
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